Physics Book - User contributions [en]

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-25T00:06:39Z

Adambarletta:

Created by Adam Barletta - Spring 2022

== Introduction ==
As you may have already learned, the Single Particle in a Box problem is a greatly intuitive way to begin to understand some of the major concepts connecting classical mechanics and quantum mechanics. It is recommended that you first begin by analyzing the "Infinite Well" solution prior to this "Semi-Infinite Well". Once you have a solid understanding of the concepts utilized in that example, you will find yourself better able to understand this specific example and explore the nuances that come with it. Like the "Infinite Well" example, we begin with a particle in a well where the potential energy where <math> x < 0 </math> is infinity. We will call the region of infinite potential "Region I". The region where <math> 0 < x < L </math> (<math> L </math> is any arbitrary distance from <math> x </math>) possesses a potential energy of <math> 0 </math> ("Region II"), and the area <math> x > L </math> has an unchanging potential equal to the positive constant <math> V_{0} </math> ("Region III").

==Background==

Of course, we will start with Schrödinger's Equation:
<math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+V\Psi=E\Psi </math>.
This specific form of Schrödinger's Equation is known as a "1-Dimensional Time-Independent Schrödinger Equation" due to its singular spacial dimension and its lack of dependency on time. The reason for the singular spacial dimension is for simplicity. The equation does not depend on time due to the the potential regions being only dependent on position, allowing us to ignore time for now. The variable <math> \hbar </math> represents Planck's constant <math> h </math> (<math> 6.62607015\times 10^{-34} \frac{\text{m}^{2}\text{kg}}{\text{s}} </math>) divided by <math> 2\pi </math>, <math> m </math> represents the mass of the singular particle, <math> \Psi </math> represents the wave function we are trying to find, <math> V </math> represents the potential energy, and <math> E </math> represents the energy of the particle. Due to the piece-wise nature of our potential, we will solve this equation in each region separately at first.

==Method to Solve==

The procedure we will use to solve this problem will go as follows:
#The general solution to the Schrödinger Equation will be found in all regions
#Each region extending to positive or negative infinity will be made finite to comply with the requirements of a proper wave function
#The solution will be made continuous at all boundaries and made differentiable at non-infinite potential boundaries
#Analyze and find intuitive concepts

Following each of these steps is crucial for finding an applicable solution. Although this is an impossible real world scenario, following these steps will assure that this example is intuitive and beneficial to our understanding of quantum wave functions.

==Solution==

===Step 1: General Solutions in Each Region===

====Region I====
First substituting the given infinite potential into the Schrödinger Equation: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(\infty) \Psi=E\Psi </math> <br /> <br />Thinking back on the "Infinite Well" solution, we can intuitively find that in order for this equation to make sense, the wave function:<br /> <br /> <math> \Psi_{I}=0 </math> <br /> <br />The reason for this is due to the fact that a theoretically infinite potential would be not allow for a particle to exist in given it has a finite energy. The likely hood of the particle existing in this region is represented by the wave function, so given that it would be impossible we can infer that the wave function would equal zero in this specific region.

====Region II====
Again, we will first substitute the given potential, zero, into the Schrödinger Equation:<br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(0) \Psi=E\Psi </math> <br /> <br />
This equation then turns into: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=E\Psi </math> <br /> <br / > Due to the constants in this equations, we can determine that the general solution of the wave function can be written using trigonometric functions or exponentials. For this example we will use the trigonometric general solution: <br /> <br /> <math> \Psi_{II}=Asin(kx)+Bcos(kx) </math> <br /> <br />Where <math> A </math>, <math> B </math>, and <math> k </math> are arbitrary constants. <math> k </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{II} </math>, gives: <br /> <br /><math> \frac{\hbar ^{2}k^{2}}{2m}[Asin(kx)+Bsin(kx)]=E[Asin(kx)+Bsin(kx)] </math> <br /> <br />Manipulating to solve for <math> k </math>: <br /m> <br /m> <math> k=\frac{\sqrt{2mE}}{\hbar } </math>

====Region III====
When solving Schrödinger Equation with a finite constant potential, <math> V_{0} </math>, the wave function will differ given the scenario that the particles energy, <math> E > V_{0} </math> or <math> E < V_{0} </math>. Given that this is a "Semi-Infinite Well" problem, we will maintain the "well" scenario by assuming '''<math> E < V_{0} </math>'''. I personally implore you attempt to solve this problem given <math> E > V_{0} </math> and discover how these answers differ. <br /><br /> Once more, substituting our given potential <math> V_{0} </math> into the Schrödinger Equation: <br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(V_{0}) \Psi=E\Psi </math> <br /> <br />Keeping in mind that <math> E < V_{0} </math>, the equation can be rewritten in the following form: <br /><br /> <math> \frac{\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=(V_{0}-E)\Psi </math> <br /><br /> Similar to "Region II", we now find ourselves with a simple differential equation with constants. Unlike "Region II", we will instead chose to utilize the exponential form of the wave equation given that this region is classically forbidden to a particle with energy <math> E < V_{0} </math>. Our wave function must then take the form: <br /m> <br /> <math> \Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />Where <math> C </math>, <math> D </math>, and <math> k' </math> are arbitrary constants. <math> k' </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{III} </math>, gives: <br /> <br /> <math> -\frac{\hbar ^{2}k'^{2}}{2m}(Ce^{k'x}+De^{-k'x})=(E-V_{0})(Ce^{k'x}+De^{-k'x}) </math> <br /> <br /> Manipulating to solve for <math> k' </math>: <br /m> <br /m> <math> k'=\frac{\sqrt{2m(V_{0}-E)}}{\hbar } </math>

===Step 2: General Solutions at Positive and Negative Infinity===
Intuition tells us that the value of the wave equation cannot "blow up" at either positive or negative infinity. This is due to the fact that the particle must have a total probability of existing at any point in space of 100% (it must exist somewhere). If our wave function "blows up" then we would have an non-applicable answer where the probability of the particle existing at a given position would not fit the requirements of quantum mechanics we've confirmed with numerous experimental trials. Applying this condition only applies to "Region I" and "Region "III".

====Region I====
Given that our wave function in this region currently has a value: <br /> <br /> <math> \Psi_{I} = 0 </math> <br /> <br />It will not blow up as x approaches negative infinity. We are free to leave this region's wave function as it is.

====Region III====
Our wave function in this region is currently being described in the form:<br /> <br /><math>\Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />In order for this equation to not blow up as x approaches infinity, we must set our constant <math> C = 0 </math>. This leaves us with a decaying function:<br /> <br /><math>\Psi_{III}=De^{-k'x} </math><br /> <br />Where the wave function approaches 0 as x approaches infinity. Our wave function now fits this quantum wave function requirement.

===Step 3: Applying Boundary Conditions===
Recall that as stated earlier, this piece-wise wave function must be continuous at every point. At points where an infinite potential is not involved, it must also be differentiable. The boundaries we must adjust for are <math> x = 0 </math> and <math> x = L </math>.

====Boundary: <math> x = 0 </math>====
As stated earlier, any boundary point that involves an infinite potential must only be continuous. When <math> x < 0 </math>, potential energy equals infinity. This allows us to only need to fulfill the condition that:<br /><br /><math> \Psi_{I}(0)=\Psi_{II}(0)</math><br /><br />Substituting in the given x value for each wave equation:<br /><br /><math> 0=Asin(k(0))+Bcos(k(0)) </math><br /><math> 0 = A(0) + B(1) </math> <br /> <math> B = 0 </math><br /><br />Now knowing that our constant <math> B = 0 </math>, we can rewrite our wave equations so they fit the boundary conditions.<br /><br /><math>\Psi_{I}=0</math><br /><math>\Psi_{II}= Asin(kx)</math>

====Boundary: <math> x = L </math>====
This boundary condition lies on the border of two finite potentials, requiring us to have continuity and differentiability between our two wave functions. We will start by applying these conditions:<br /><br /><math>\Psi_{II}(L)=\Psi_{III}(L)</math><br /><math>\Psi_{II}'(L)=\Psi_{III}'(L)</math><br /><br />For mathematical simplicity we can express these equations as ratios:<br /><br /><math>\frac{\Psi_{II}'(L)}{\Psi_{II}(L)}=\frac{\Psi_{III}'(L)}{\Psi_{III}(L)}</math><br /><br />Differentiating each wave equation:<br /><br /><math>\Psi_{II}'=Akcos(kx)</math><br /><math>\Psi_{III}'=-Dk'e^{-k'x}</math><br /><br />Substituting into the ratio we created previously and plugging in our given value of x:<br /><br /><math>\frac{Akcos(kL)}{Asin(kL)}=\frac{-Dk'e^{k'L}}{De^{k'L}}</math><br /><br />Which then conveniently simplifies down to:<br /><br /><math>kcot(kL)=-k'</math><br /><br />Now that we have a clear representation of the relationship between <math> k </math> and <math> k' </math>, we can begin to better understand the implications our calculations have in the domain of quantum mechanics.

==Final Analysis==

After many rigorous calculations, we have finally created a solution for the "Semi-Infinite Well" problem. Given the regions of <math> x < 0 </math> ("Region I" with infinite potential), <math> 0 < x < L </math> ("Region II" with zero potential), and <math> x > L </math> ("Region III" with finite, postive potential <math> V_{0} </math>), Our final wave functions are as follows:<br /><br /><math>\Psi_{I}=0</math><br /><math>\Psi_{II}=Asin(kx)</math><br /><math>\Psi_{III}=De^{-k'x}</math> <br /><br /> Which consist of the following relationships:<br /><br />
#<math>k=\frac{\sqrt{2mE}}{\hbar }</math>
#<math> k'=\frac{\sqrt{2m(V_{0}-E)}}{\hbar } </math>
#<math>kcot(kL)=-k'</math>
<br /><br />Now what does all this tell us? To start, If we were to plug in our known values of <math> k </math> and <math> k' </math> in for our third listed relationship, we would notice something interesting. The equality only is true for discrete values of <math> E </math>. This relationship displays the idea of the quantization of energy! Our wave function changes at these different energy levels, and the particle can only possess the quantized energy predicted by our relationships. Secondly, you may have noticed that unlike the infinite potential region, the non-zero potential region has a decaying wave function. We know that there is no such thing as an infinite potential in the real world, so this decaying function has startling implications in the world of quantum physics. In this non-zero potential region, the particle's likelihood of existing is extremely low but never zero. If we were to reduce the width of this region and have it act more as a wall between two zero potential regions, we could possibly see the particle pop out on the other side! This is of course was not thought to be possible in classical physics. Classically, no matter how long you stare at a ball in a valley, for example, it would never miraculously roll all the way up one of the sides and escape the valley. This is not true for quantum physics! As we enter the domain of particles we observe this exact behavior and it is known as Quantum Tunneling.
<br /><br /><br />

References:<br />
Dr. John M. Standard's 2013 "The Particle in a Half-Infinite Well", url: https://www.yumpu.com/en/document/read/34504139/the-particle-in-a-half-infinite-well<br />
https://www.chegg.com/homework-help/questions-and-answers/energy-values-first-part-proble-e1-1187-ev-e2-4667-ev-q31361434<br />
Paul D'Alessandris's 2021 "Solving the 1D Semi-Infinite Square Well" url: https://phys.libretexts.org/Bookshelves/Modern_Physics/Book%3A_Spiral_Modern_Physics_(D'Alessandris)/6%3A_The_Schrodinger_Equation/6.6%3A_Solving_the_1D_Semi-Infinite_Square_Well

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T23:58:46Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T23:56:58Z

Adambarletta:

Created by Adam Barletta - Spring 2022

== Introduction ==
As you may have already learned, the Single Particle in a Box problem is a greatly intuitive way to begin to understand some of the major concepts connecting classical mechanics and quantum mechanics. It is recommended that you first begin by analyzing the "Infinite Well" solution prior to this "Semi-Infinite Well". Once you have a solid understanding of the concepts utilized in that example, you will find yourself better able to understand this specific example and explore the nuances that come with it. Like the "Infinite Well" example, we begin with a particle in a well where the potential energy where <math> x < 0 </math> is infinity. We will call the region of infinite potential "Region I". The region where <math> 0 < x < L </math> (<math> L </math> is any arbitrary distance from <math> x </math>) possesses a potential energy of <math> 0 </math> ("Region II"), and the area <math> x > L </math> has an unchanging potential equal to the positive constant <math> V_{0} </math> ("Region III").

==Background==

Of course, we will start with Schrödinger's Equation:
<math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+V\Psi=E\Psi </math>.
This specific form of Schrödinger's Equation is known as a "1-Dimensional Time-Independent Schrödinger Equation" due to its singular spacial dimension and its lack of dependency on time. The reason for the singular spacial dimension is for simplicity. The equation does not depend on time due to the the potential regions being only dependent on position, allowing us to ignore time for now. The variable <math> \hbar </math> represents Planck's constant <math> h </math> (<math> 6.62607015\times 10^{-34} \frac{\text{m}^{2}\text{kg}}{\text{s}} </math>) divided by <math> 2\pi </math>, <math> m </math> represents the mass of the singular particle, <math> \Psi </math> represents the wave function we are trying to find, <math> V </math> represents the potential energy, and <math> E </math> represents the energy of the particle. Due to the piece-wise nature of our potential, we will solve this equation in each region separately at first.

==Method to Solve==

The procedure we will use to solve this problem will go as follows:
#The general solution to the Schrödinger Equation will be found in all regions
#Each region extending to positive or negative infinity will be made finite to comply with the requirements of a proper wave function
#The solution will be made continuous at all boundaries and made differentiable at non-infinite potential boundaries
#Analyze and find intuitive concepts

Following each of these steps is crucial for finding an applicable solution. Although this is an impossible real world scenario, following these steps will assure that this example is intuitive and beneficial to our understanding of quantum wave functions.

==Solution==

===Step 1: General Solutions in Each Region===

====Region I====
First substituting the given infinite potential into the Schrödinger Equation: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(\infty) \Psi=E\Psi </math> <br /> <br />Thinking back on the "Infinite Well" solution, we can intuitively find that in order for this equation to make sense, the wave function:<br /> <br /> <math> \Psi_{I}=0 </math> <br /> <br />The reason for this is due to the fact that a theoretically infinite potential would be not allow for a particle to exist in given it has a finite energy. The likely hood of the particle existing in this region is represented by the wave function, so given that it would be impossible we can infer that the wave function would equal zero in this specific region.

====Region II====
Again, we will first substitute the given potential, zero, into the Schrödinger Equation:<br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(0) \Psi=E\Psi </math> <br /> <br />
This equation then turns into: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=E\Psi </math> <br /> <br / > Due to the constants in this equations, we can determine that the general solution of the wave function can be written using trigonometric functions or exponentials. For this example we will use the trigonometric general solution: <br /> <br /> <math> \Psi_{II}=Asin(kx)+Bcos(kx) </math> <br /> <br />Where <math> A </math>, <math> B </math>, and <math> k </math> are arbitrary constants. <math> k </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{II} </math>, gives: <br /> <br /><math> \frac{\hbar ^{2}k^{2}}{2m}[Asin(kx)+Bsin(kx)]=E[Asin(kx)+Bsin(kx)] </math> <br /> <br />Manipulating to solve for <math> k </math>: <br /m> <br /m> <math> k=\frac{\sqrt{2mE}}{\hbar } </math>

====Region III====
When solving Schrödinger Equation with a finite constant potential, <math> V_{0} </math>, the wave function will differ given the scenario that the particles energy, <math> E > V_{0} </math> or <math> E < V_{0} </math>. Given that this is a "Semi-Infinite Well" problem, we will maintain the "well" scenario by assuming '''<math> E < V_{0} </math>'''. I personally implore you attempt to solve this problem given <math> E > V_{0} </math> and discover how these answers differ. <br /><br /> Once more, substituting our given potential <math> V_{0} </math> into the Schrödinger Equation: <br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(V_{0}) \Psi=E\Psi </math> <br /> <br />Keeping in mind that <math> E < V_{0} </math>, the equation can be rewritten in the following form: <br /><br /> <math> \frac{\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=(V_{0}-E)\Psi </math> <br /><br /> Similar to "Region II", we now find ourselves with a simple differential equation with constants. Unlike "Region II", we will instead chose to utilize the exponential form of the wave equation given that this region is classically forbidden to a particle with energy <math> E < V_{0} </math>. Our wave function must then take the form: <br /m> <br /> <math> \Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />Where <math> C </math>, <math> D </math>, and <math> k' </math> are arbitrary constants. <math> k' </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{III} </math>, gives: <br /> <br /> <math> -\frac{\hbar ^{2}k'^{2}}{2m}(Ce^{k'x}+De^{-k'x})=(E-V_{0})(Ce^{k'x}+De^{-k'x}) </math> <br /> <br /> Manipulating to solve for <math> k' </math>: <br /m> <br /m> <math> k'=\frac{\sqrt{2m(V_{0}-E)}}{\hbar } </math>

===Step 2: General Solutions at Positive and Negative Infinity===
Intuition tells us that the value of the wave equation cannot "blow up" at either positive or negative infinity. This is due to the fact that the particle must have a total probability of existing at any point in space of 100% (it must exist somewhere). If our wave function "blows up" then we would have an non-applicable answer where the probability of the particle existing at a given position would not fit the requirements of quantum mechanics we've confirmed with numerous experimental trials. Applying this condition only applies to "Region I" and "Region "III".

====Region I====
Given that our wave function in this region currently has a value: <br /> <br /> <math> \Psi_{I} = 0 </math> <br /> <br />It will not blow up as x approaches negative infinity. We are free to leave this region's wave function as it is.

====Region III====
Our wave function in this region is currently being described in the form:<br /> <br /><math>\Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />In order for this equation to not blow up as x approaches infinity, we must set our constant <math> C = 0 </math>. This leaves us with a decaying function:<br /> <br /><math>\Psi_{III}=De^{-k'x} </math><br /> <br />Where the wave function approaches 0 as x approaches infinity. Our wave function now fits this quantum wave function requirement.

===Step 3: Applying Boundary Conditions===
Recall that as stated earlier, this piece-wise wave function must be continuous at every point. At points where an infinite potential is not involved, it must also be differentiable. The boundaries we must adjust for are <math> x = 0 </math> and <math> x = L </math>.

====Boundary: <math> x = 0 </math>====
As stated earlier, any boundary point that involves an infinite potential must only be continuous. When <math> x < 0 </math>, potential energy equals infinity. This allows us to only need to fulfill the condition that:<br /><br /><math> \Psi_{I}(0)=\Psi_{II}(0)</math><br /><br />Substituting in the given x value for each wave equation:<br /><br /><math> 0=Asin(k(0))+Bcos(k(0)) </math><br /><math> 0 = A(0) + B(1) </math> <br /> <math> B = 0 </math><br /><br />Now knowing that our constant <math> B = 0 </math>, we can rewrite our wave equations so they fit the boundary conditions.<br /><br /><math>\Psi_{I}=0</math><br /><math>\Psi_{II}= Asin(kx)</math>

====Boundary: <math> x = L </math>====
This boundary condition lies on the border of two finite potentials, requiring us to have continuity and differentiability between our two wave functions. We will start by applying these conditions:<br /><br /><math>\Psi_{II}(L)=\Psi_{III}(L)</math><br /><math>\Psi_{II}'(L)=\Psi_{III}'(L)</math><br /><br />For mathematical simplicity we can express these equations as ratios:<br /><br /><math>\frac{\Psi_{II}'(L)}{\Psi_{II}(L)}=\frac{\Psi_{III}'(L)}{\Psi_{III}(L)}</math><br /><br />Differentiating each wave equation:<br /><br /><math>\Psi_{II}'=Akcos(kx)</math><br /><math>\Psi_{III}'=-Dk'e^{-k'x}</math><br /><br />Substituting into the ratio we created previously and plugging in our given value of x:<br /><br /><math>\frac{Akcos(kL)}{Asin(kL)}=\frac{-Dk'e^{k'L}}{De^{k'L}}</math><br /><br />Which then conveniently simplifies down to:<br /><br /><math>kcot(kL)=-k'</math><br /><br />Now that we have a clear representation of the relationship between <math> k </math> and <math> k' </math>, we can begin to better understand the implications our calculations have in the domain of quantum mechanics.

==Final Analysis==

After many rigorous calculations, we have finally created a solution for the "Semi-Infinite Well" problem. Given the regions of <math> x < 0 </math> ("Region I" with infinite potential), <math> 0 < x < L </math> ("Region II" with zero potential), and <math> x > L </math> ("Region III" with finite, postive potential <math> V_{0} </math>), Our final wave functions are as follows:<br /><br /><math>\Psi_{I}=0</math><br /><math>\Psi_{II}=Asin(kx)</math><br /><math>\Psi_{III}=De^{-k'x}</math> <br /><br /> Which consist of the following relationships:<br /><br />
#<math>k=\frac{\sqrt{2mE}}{\hbar }</math>
#<math> k'=\frac{\sqrt{2m(V_{0}-E)}}{\hbar } </math>
#<math>kcot(kL)=-k'</math>
<br /><br />Now what does all this tell us? To start, If we were to plug in our known values of <math> k </math> and <math> k' </math> in for our third listed relationship, we would notice something interesting. The equality only is true for discrete values of <math> E </math>. This relationship displays the idea of the quantization of energy! Our wave function changes at these different energy levels, and the particle can only possess the quantized energy predicted by our relationships. Secondly, you may have noticed that unlike the infinite potential region, the non-zero potential region has a decaying wave function. We know that there is no such thing as an infinite potential in the real world, so this decaying function has startling implications in the world of quantum physics. In this non-zero potential region, the particles likely hood of existing is extremely low but never zero. If we were to reduce the width of this region and have it act more as a wall between two zero potential regions, we could possibly see the particle pop out on the other side! This is of course was not thought to be possible in classical physics. Classically, no matter how long you stare at a ball in a valley, for example, it would never miraculously roll all the way up one of the sides and escape the valley. This is not true for quantum physics! As we enter the domain of particles we observe this exact behavior and it is known as Quantum Tunneling.

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T23:02:07Z

Adambarletta:

Created by Adam Barletta - Spring 2022

== Introduction ==
As you may have already learned, the Single Particle in a Box problem is a greatly intuitive way to begin to understand some of the major concepts connecting classical mechanics and quantum mechanics. It is recommended that you first begin by analyzing the "Infinite Well" solution prior to this "Semi-Infinite Well". Once you have a solid understanding of the concepts utilized in that example, you will find yourself better able to understand this specific example and explore the nuances that come with it. Like the "Infinite Well" example, we begin with a particle in a well where the potential energy where <math> x < 0 </math> is infinity. We will call the region of infinite potential "Region I". The region where <math> 0 < x < L </math> (<math> L </math> is any arbitrary distance from <math> x </math>) possesses a potential energy of <math> 0 </math> ("Region II"), and the area <math> x > L </math> has an unchanging potential equal to the positive constant <math> V_{0} </math> ("Region III").

==Background==

Of course, we will start with Schrödinger's Equation:
<math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+V\Psi=E\Psi </math>.
This specific form of Schrödinger's Equation is known as a "1-Dimensional Time-Independent Schrödinger Equation" due to its singular spacial dimension and its lack of dependency on time. The reason for the singular spacial dimension is for simplicity. The equation does not depend on time due to the the potential regions being only dependent on position, allowing us to ignore time for now. The variable <math> \hbar </math> represents Planck's constant <math> h </math> (<math> 6.62607015\times 10^{-34} \frac{\text{m}^{2}\text{kg}}{\text{s}} </math>) divided by <math> 2\pi </math>, <math> m </math> represents the mass of the singular particle, <math> \Psi </math> represents the wave function we are trying to find, <math> V </math> represents the potential energy, and <math> E </math> represents the energy of the particle. Due to the piece-wise nature of our potential, we will solve this equation in each region separately at first.

==Method to Solve==

The procedure we will use to solve this problem will go as follows:
#The general solution to the Schrödinger Equation will be found in all regions
#Each region extending to positive or negative infinity will be made finite to comply with the requirements of a proper wave function
#The solution will be made continuous at all boundaries and made differentiable at non-infinite potential boundaries
#Assure the equation is properly normalized

Following each of these steps is crucial for finding an applicable solution. Although this is an impossible real world scenario, following these steps will assure that this example is intuitive and beneficial to our understanding of quantum wave functions.

==Solution==

===Step 1: General Solutions in Each Region===

====Region I====
First substituting the given infinite potential into the Schrödinger Equation: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(\infty) \Psi=E\Psi </math> <br /> <br />Thinking back on the "Infinite Well" solution, we can intuitively find that in order for this equation to make sense, the wave function:<br /> <br /> <math> \Psi_{I}=0 </math> <br /> <br />The reason for this is due to the fact that a theoretically infinite potential would be not allow for a particle to exist in given it has a finite energy. The likely hood of the particle existing in this region is represented by the wave function, so given that it would be impossible we can infer that the wave function would equal zero in this specific region.

====Region II====
Again, we will first substitute the given potential, zero, into the Schrödinger Equation:<br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(0) \Psi=E\Psi </math> <br /> <br />
This equation then turns into: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=E\Psi </math> <br /> <br / > Due to the constants in this equations, we can determine that the general solution of the wave function can be written using trigonometric functions or exponentials. For this example we will use the trigonometric general solution: <br /> <br /> <math> \Psi_{II}=Asin(kx)+Bcos(kx) </math> <br /> <br />Where <math> A </math>, <math> B </math>, and <math> k </math> are arbitrary constants. <math> k </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{II} </math>, gives: <br /> <br /><math> \frac{\hbar ^{2}k^{2}}{2m}[Asin(kx)+Bsin(kx)]=E[Asin(kx)+Bsin(kx)] </math> <br /> <br />Manipulating to solve for <math> k </math>: <br /m> <br /m> <math> k=\frac{\sqrt{2mE}}{\hbar } </math>

====Region III====
When solving Schrödinger Equation with a finite constant potential, <math> V_{0} </math>, the wave function will differ given the scenario that the particles energy, <math> E > V_{0} </math> or <math> E < V_{0} </math>. Given that this is a "Semi-Infinite Well" problem, we will maintain the "well" scenario by assuming '''<math> E < V_{0} </math>'''. I personally implore you attempt to solve this problem given <math> E > V_{0} </math> and discover how these answers differ. <br /><br /> Once more, substituting our given potential <math> V_{0} </math> into the Schrödinger Equation: <br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(V_{0}) \Psi=E\Psi </math> <br /> <br />Keeping in mind that <math> E < V_{0} </math>, the equation can be rewritten in the following form: <br /><br /> <math> \frac{\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=(V_{0}-E)\Psi </math> <br /><br /> Similar to "Region II", we now find ourselves with a simple differential equation with constants. Unlike "Region II", we will instead chose to utilize the exponential form of the wave equation given that this region is classically forbidden to a particle with energy <math> E < V_{0} </math>. Our wave function must then take the form: <br /m> <br /> <math> \Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />Where <math> C </math>, <math> D </math>, and <math> k' </math> are arbitrary constants. <math> k' </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{III} </math>, gives: <br /> <br /> <math> -\frac{\hbar ^{2}k'^{2}}{2m}(Ce^{k'x}+De^{-k'x})=(E-V_{0})(Ce^{k'x}+De^{-k'x}) </math> <br /> <br /> Manipulating to solve for <math> k' </math>: <br /m> <br /m> <math> k'=\frac{\sqrt{2m(V_{0}-E)}}{\hbar } </math>

===Step 2: General Solutions at Positive and Negative Infinity===
Intuition tells us that the value of the wave equation cannot "blow up" at either positive or negative infinity. This is due to the fact that the particle must have a total probability of existing at any point in space of 100% (it must exist somewhere). If our wave function "blows up" then we would have an non-applicable answer where the probability of the particle existing at a given position would not fit the requirements of quantum mechanics we've confirmed with numerous experimental trials. Applying this condition only applies to "Region I" and "Region "III".

====Region I====
Given that our wave function in this region currently has a value: <br /> <br /> <math> \Psi_{I} = 0 </math> <br /> <br />It will not blow up as x approaches negative infinity. We are free to leave this region's wave function as it is.

====Region III====
Our wave function in this region is currently being described in the form:<br /> <br /><math>\Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />In order for this equation to not blow up as x approaches infinity, we must set our constant <math> C = 0 </math>. This leaves us with a decaying function:<br /> <br /><math>\Psi_{III}=De^{-k'x} </math><br /> <br />Where the wave function approaches 0 as x approaches infinity. Our wave function now fits this quantum wave function requirement.

===Step 3: Applying Boundary Conditions===
Recall that as stated earlier, this piece-wise wave function must be continuous at every point. At points where an infinite potential is not involved, it must also be differentiable. The boundaries we must adjust for are <math> x = 0 </math> and <math> x = L </math>.

====Boundary: <math> x = 0 </math>====
As stated earlier, any boundary point that involves an infinite potential must only be continuous. When <math> x < 0 </math>, potential energy equals infinity. This allows us to only need to fulfill the condition that:<br /><br /><math> \Psi_{I}(0)=\Psi_{II}(0)</math><br /><br />Substituting in the given x value for each wave equation:<br /><br /><math> 0=Asin(k(0))+Bcos(k(0)) </math><br /><math> 0 = A(0) + B(1) </math> <br /> <math> B = 0 </math><br /><br />Now knowing that our constant <math> B = 0 </math>, we can rewrite our wave equations so they fit the boundary conditions.<br /><br /><math>\Psi_{I}=0</math><br /><math>\Psi_{II}= Asin(kx)</math>

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T22:46:09Z

Adambarletta:

Claimed by Adam Barletta 4/21/22

== Introduction ==
As you may have already learned, the Single Particle in a Box problem is a greatly intuitive way to begin to understand some of the major concepts connecting classical mechanics and quantum mechanics. It is recommended that you first begin by analyzing the "Infinite Well" solution prior to this "Semi-Infinite Well". Once you have a solid understanding of the concepts utilized in that example, you will find yourself better able to understand this specific example and explore the nuances that come with it. Like the "Infinite Well" example, we begin with a particle in a well where the potential energy where <math> x < 0 </math> is infinity. We will call the region of infinite potential "Region I". The region where <math> 0 < x < L </math> (<math> L </math> is any arbitrary distance from <math> x </math>) possesses a potential energy of <math> 0 </math> ("Region II"), and the area <math> x > L </math> has an unchanging potential equal to the positive constant <math> V_{0} </math> ("Region III").

==Background==

Of course, we will start with Schrödinger's Equation:
<math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+V\Psi=E\Psi </math>.
This specific form of Schrödinger's Equation is known as a "1-Dimensional Time-Independent Schrödinger Equation" due to its singular spacial dimension and its lack of dependency on time. The reason for the singular spacial dimension is for simplicity. The equation does not depend on time due to the the potential regions being only dependent on position, allowing us to ignore time for now. The variable <math> \hbar </math> represents Planck's constant <math> h </math> (<math> 6.62607015\times 10^{-34} \frac{\text{m}^{2}\text{kg}}{\text{s}} </math>) divided by <math> 2\pi </math>, <math> m </math> represents the mass of the singular particle, <math> \Psi </math> represents the wave function we are trying to find, <math> V </math> represents the potential energy, and <math> E </math> represents the energy of the particle. Due to the piece-wise nature of our potential, we will solve this equation in each region separately at first.

==Method to Solve==

The procedure we will use to solve this problem will go as follows:
#The general solution to the Schrödinger Equation will be found in all regions
#Each region extending to positive or negative infinity will be made finite to comply with the requirements of a proper wave function
#The solution will be made continuous at all boundaries and made differentiable at non-infinite potential boundaries
#Assure the equation is properly normalized

Following each of these steps is crucial for finding an applicable solution. Although this is an impossible real world scenario, following these steps will assure that this example is intuitive and beneficial to our understanding of quantum wave functions.

==Solution==

===Step 1: General Solutions in Each Region===

====Region I====
First substituting the given infinite potential into the Schrödinger Equation: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(\infty) \Psi=E\Psi </math> <br /> <br />Thinking back on the "Infinite Well" solution, we can intuitively find that in order for this equation to make sense, the wave function:<br /> <br /> <math> \Psi_{I}=0 </math> <br /> <br />The reason for this is due to the fact that a theoretically infinite potential would be not allow for a particle to exist in given it has a finite energy. The likely hood of the particle existing in this region is represented by the wave function, so given that it would be impossible we can infer that the wave function would equal zero in this specific region.

====Region II====
Again, we will first substitute the given potential, zero, into the Schrödinger Equation:<br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(0) \Psi=E\Psi </math> <br /> <br />
This equation then turns into: <br /> <br /> <math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=E\Psi </math> <br /> <br / > Due to the constants in this equations, we can determine that the general solution of the wave function can be written using trigonometric functions or exponentials. For this example we will use the trigonometric general solution: <br /> <br /> <math> \Psi_{II}=Asin(kx)+Bcos(kx) </math> <br /> <br />Where <math> A </math>, <math> B </math>, and <math> k </math> are arbitrary constants. <math> k </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{II} </math>, gives: <br /> <br /><math> \frac{\hbar ^{2}k^{2}}{2m}[Asin(kx)+Bsin(kx)]=E[Asin(kx)+Bsin(kx)] </math> <br /> <br />Manipulating to solve for <math> k </math>: <br /m> <br /m> <math> k=\frac{\sqrt{2mE}}{\hbar } </math>

====Region III====
When solving Schrödinger Equation with a finite constant potential, <math> V_{0} </math>, the wave function will differ given the scenario that the particles energy, <math> E > V_{0} </math> or <math> E < V_{0} </math>. Given that this is a "Semi-Infinite Well" problem, we will maintain the "well" scenario by assuming '''<math> E < V_{0} </math>'''. I personally implore you attempt to solve this problem given <math> E > V_{0} </math> and discover how these answers differ. <br /><br /> Once more, substituting our given potential <math> V_{0} </math> into the Schrödinger Equation: <br /> <br /><math> \frac{-\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}+(V_{0}) \Psi=E\Psi </math> <br /> <br />Keeping in mind that <math> E < V_{0} </math>, the equation can be rewritten in the following form: <br /><br /> <math> \frac{\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi}{\partial x^{2}}=(V_{0}-E)\Psi </math> <br /><br /> Similar to "Region II", we now find ourselves with a simple differential equation with constants. Unlike "Region II", we will instead chose to utilize the exponential form of the wave equation given that this region is classically forbidden to a particle with energy <math> E < V_{0} </math>. Our wave function must then take the form: <br /m> <br /> <math> \Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />Where <math> C </math>, <math> D </math>, and <math> k' </math> are arbitrary constants. <math> k' </math> can be solved for by substituting our general solution back into the original Schrödinger Equation, taking the second derivative, and preforming some algebraic manipulations. The second derivative of the Schrödinger Equation with the wave function we found, <math> \Psi_{III} </math>, gives: <br /> <br /> <math> -\frac{\hbar ^{2}k'^{2}}{2m}(Ce^{k'x}+De^{-k'x})=(E-V_{0})(Ce^{k'x}+De^{-k'x}) </math> <br /> <br /> Manipulating to solve for <math> k' </math>: <br /m> <br /m> <math> k'=\frac{\sqrt{2m(V_{0}-E)}}{\hbar } </math>

===Step 2: General Solutions at Positive and Negative Infinity===
Intuition tells us that the value of the wave equation cannot "blow up" at either positive or negative infinity. This is due to the fact that the particle must have a total probability of existing at any point in space of 100% (it must exist somewhere). If our wave function "blows up" then we would have an non-applicable answer where the probability of the particle existing at a given position would not fit the requirements of quantum mechanics we've confirmed with numerous experimental trials. Applying this condition only applies to "Region I" and "Region "III".

====Region I====
Given that our wave function in this region currently has a value: <br /> <br /> <math> \Psi_{I} = 0 </math> <br /> <br />It will not blow up as x approaches negative infinity. We are free to leave this region's wave function as it is.

====Region III====
Our wave function in this region is currently being described in the form:<br /> <br /><math>\Psi_{III}=Ce^{k'x}+De^{-k'x} </math><br /> <br />In order for this equation to not blow up as x approaches infinity, we must set our constant <math> C = 0 </math>. This leaves us with a decaying function:<br /> <br /><math>\Psi_{III}=De^{-k'x} </math><br /> <br />Where the wave function approaches 0 as x approaches infinity. Our wave function now fits this quantum wave function requirement.

===Step 3: Applying Boundary Conditions===
Recall that as stated earlier, this piece-wise wave function must be continuous at every point. At points where an infinite potential is not involved, it must also be differentiable. The boundaries we must adjust for are <math> x = 0 </math> and <math> x = L </math>.

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T22:36:35Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T22:17:47Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T22:08:05Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T22:06:09Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T21:58:21Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T21:53:03Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T21:49:13Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T21:43:10Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T21:15:57Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T20:56:20Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T20:48:37Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T20:12:13Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T20:11:38Z

Adambarletta:

File:Potential-X-Graph.png

2022-04-24T20:00:52Z

Adambarletta: Credit: https://www.chegg.com/homework-help/questions-and-answers/energy-values-first-part-proble-e1-1187-ev-e2-4667-ev-q31361434

== Summary ==
Credit: https://www.chegg.com/homework-help/questions-and-answers/energy-values-first-part-proble-e1-1187-ev-e2-4667-ev-q31361434

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-24T19:56:09Z

Adambarletta:

Solution for a Single Particle in a Semi-Infinite Quantum Well

2022-04-22T01:52:47Z

Adambarletta: Created page with "Claimed by Adam Barletta 4/21/22"

Claimed by Adam Barletta 4/21/22

Main Page

2022-04-22T01:52:08Z

Adambarletta: /* Schrödinger Equation */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
<h1><strong>Alayna Baker Spring 2020</strong></h1>
[[File:Hall Effect 1.jpg]]
[[File:Hall Effect 2.jpg]]

*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

]]]====Motional EMF====

<div class="mw-collapsible-content">
*[[Motional Emf]]
<h1><strong>Adeline Boswell Fall 2019</strong></h1>
[[File:Motional EMF Example.jpg]]

*[[Motional Emf using Faraday's Law]]
</div>
</div>http://www.physicsbook.gatech.edu/Special:RecentChangesLinked/Main_Page

<div class="toccolours mw-collapsible mw-collapsed">

If you have a bar attached to two rails, and the rails are connected by a resistor, you have effectively created a circuit. As the bar moves, it creates an "electromotive force"

[[File:MotEMFCR.jpg]]

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[Solution for a Single Free Particle]]
<div class="mw-collapsible-content">
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
<div class="mw-collapsible-content"></div>
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
*[[Molecules]]
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
*[[Application of Statistics in Physics]]
*[[Temperature & Entropy]]
</div>
<div class="mw-collapsible-content">

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Main Page

2022-04-22T01:50:26Z

Adambarletta: /* Schrödinger Equation */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
<h1><strong>Alayna Baker Spring 2020</strong></h1>
[[File:Hall Effect 1.jpg]]
[[File:Hall Effect 2.jpg]]

*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

]]]====Motional EMF====

<div class="mw-collapsible-content">
*[[Motional Emf]]
<h1><strong>Adeline Boswell Fall 2019</strong></h1>
[[File:Motional EMF Example.jpg]]

*[[Motional Emf using Faraday's Law]]
</div>
</div>http://www.physicsbook.gatech.edu/Special:RecentChangesLinked/Main_Page

<div class="toccolours mw-collapsible mw-collapsed">

If you have a bar attached to two rails, and the rails are connected by a resistor, you have effectively created a circuit. As the bar moves, it creates an "electromotive force"

[[File:MotEMFCR.jpg]]

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[Solution for a Single Free Particle]]
<div class="mw-collapsible-content">
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
<div class="mw-collapsible-content"></div>
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well - Barletta]]

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
*[[Molecules]]
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
*[[Application of Statistics in Physics]]
*[[Temperature & Entropy]]
</div>
<div class="mw-collapsible-content">

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>