Particle in a 1-Dimensional box - Revision history

Eathan: /* See also */

2022-04-24T02:54:23Z

Eathan: /* History */

2022-04-24T02:53:25Z

History

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	==History==		==History==

	~~Put this idea~~ in ~~historical context~~. ~~Give~~ the ~~reader~~ the ~~Who~~, ~~What~~, ~~When~~, ~~Where~~, ~~and Why~~.		Erwin Schrodinger was born on August 12th, 1887 in Vienna. He would be most well known for his work in developing his namesake equation that provides a way to calculate the wave function of a quantum system and predict how it will evolve with time. He did this after scientist Peter Debye listened to a lecture given by Schrodinger concerning De Broglie's wave-particle duality theory. Debye did not agree with this De Broglie's theory since it showed the existence of matter waves, but not the equation for the wave itself. At the time, there was no equation whose solution would describe the waves created by De Broglie's theory. Erwin Schrodinger would dedicate his work to solving this problem, and by January 1926, he published his paper "Quantization as an Eigenvalue Problem" which displays his namesake equation. This paper correctly gave the energy values for a hydrogen atom for time-independent systems. Over the following year, he would publish more papers furthering his equation by solving the quantum harmonic oscillator among other accomplishments.

	== See also ==		== See also ==

Eathan: /* Connectedness */

2022-04-24T02:38:51Z

Connectedness

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	This topic of being able to solve a particle in a 1-Dimensional box is directly related to how I should approach, understand, and appreciate physics problems in the future. This idea lays the groundwork for future endeavors into Schrodinger's Equation when I take quantum mechanics later for my major.		This topic of being able to solve a particle in a 1-Dimensional box is directly related to how I should approach, understand, and appreciate physics problems in the future. This idea lays the groundwork for future endeavors into Schrodinger's Equation when I take quantum mechanics later for my major.
	#Is there an interesting industrial application?		#Is there an interesting industrial application?
			While solving the equation itself does not lead to many interesting industrial applications, the solutions of Schrodinger's Equations lend their way to explaining quantum tunneling. Quantum tunneling can be applied in many different ways, including in Scanning-Tunneling Microscopes (STM), radioactive decay, and Potassium-Argon dating.

	==History==		==History==

Eathan: /* Connectedness */

2022-04-24T02:37:08Z

Connectedness

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	==Connectedness==		==Connectedness==
	#How is this topic connected to something that you are interested in?		#How is this topic connected to something that you are interested in?
			This topic of a particle in a 1-Dimensional Box is interesting to me since it shows the direct implications of Schrodinger's Equation. Being able to understand and solve what most consider a cornerstone for quantum physics is part of the reason why I love physics so much. Not only am I able to solve this equation for many different cases, but I'm also able to understand and interpret the results. I can see how the equation lends its way to how energy must be quantized, or how it directly implies Heisenberg's uncertainty principle.
	#How is it connected to your major?		#How is it connected to your major?
			This topic of being able to solve a particle in a 1-Dimensional box is directly related to how I should approach, understand, and appreciate physics problems in the future. This idea lays the groundwork for future endeavors into Schrodinger's Equation when I take quantum mechanics later for my major.
	#Is there an interesting industrial application?		#Is there an interesting industrial application?

Eathan: /* The General Infinite Potential Well */

2022-04-23T08:45:10Z

The General Infinite Potential Well

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	<math>\psi(-frac{a}{2}) = Acos(\frac{ka}{2}) - Bsin(\frac{ka}{2}) = 0 </math>		<math>\psi(-frac{a}{2}) = Acos(\frac{ka}{2}) - Bsin(\frac{ka}{2}) = 0 </math>

	Adding these two wave functions together yields the following results: <math>2Acos(\frac{ka}{2}) </math>. This implies that <math>k = \frac{2\pi(n + \frac{1}{2})}{a}{/math}.		Adding these two wave functions together yields the following results: <math>2Acos(\frac{ka}{2}) </math>. This implies that <math>k = \frac{2\pi(n + \frac{1}{2})}{a}</math>.

	Subtracting these two wave functions yields the following results: <math>2Bsin(\frac{ka}{2})</math>. This implies that <math>k = \frac{2\pi n}{a}</math>		Subtracting these two wave functions yields the following results: <math>2Bsin(\frac{ka}{2})</math>. This implies that <math>k = \frac{2\pi n}{a}</math>.

	We can keep both of these solutions if we allow the following piece-wise function to exist:		We can keep both of these solutions if we allow the following piece-wise function to exist:
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	\end{array} \right.</math>		\end{array} \right.</math>

			This is the more general solution for the infinite potential well as we now have both triginometric terms available in our wave function.

	==Connectedness==		==Connectedness==

Eathan at 08:43, 23 April 2022

2022-04-23T08:43:19Z

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 Here we demonstrate the exponential behavior of the wave function as it enters <math>x \geq L </math>. As seen, the wave function can exist beyond the barrier due to quantum tunneling.
 ==Examples==
-==The General Infinite Potential Well==
+===The General Infinite Potential Well===
 Consider the following quantum system as shown below
 [[File:Abs_potential.png]]
 ==Connectedness==

Eathan at 08:28, 23 April 2022

2022-04-23T08:28:55Z

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	Here we demonstrate the exponential behavior of the wave function as it enters <math>x \geq L </math>. As seen, the wave function can exist beyond the barrier due to quantum tunneling.		Here we demonstrate the exponential behavior of the wave function as it enters <math>x \geq L </math>. As seen, the wave function can exist beyond the barrier due to quantum tunneling.
	==Examples==		==Examples==
			==The General Infinite Potential Well==
			Consider the following quantum system as shown below

	~~Be sure to show all steps in your solution and include diagrams whenever possible~~		[[File:Abs_potential.png]]

	~~===Simple===~~
	~~===Middling===~~
	~~===Difficult===~~

	==Connectedness==		==Connectedness==

Eathan at 08:09, 23 April 2022

2022-04-23T08:09:28Z

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	We will first consider the following scenario:		We will first consider the following scenario:
	The particle in a 1-Dimensional box is a quantum system in which a particle is bounded in a well with infinite energy at the barrier. Since the barriers of the well are infinite, there should be a 0% chance of finding the particle outside the well. We will apply this boundary condition later as we derive the solution to the particle in a 1-Dimensional box		The particle in a 1-Dimensional box is a quantum system in which a particle is bounded in a well with infinite energy at the barrier. Since the barriers of the well are infinite, there should be a 0% chance of finding the particle outside the well. We will apply this boundary condition later as we derive the solution to the particle in a 1-Dimensional box

	[[File:Particle_in_a_boxx.png]]		[[File:Particle_in_a_boxx.png]]

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	===Infinite Potential (Computational Model)===		===Infinite Potential (Computational Model)===
	Here, we can visualize the wave function and its associated probability density for various quantum levels. As you can see, the wave function changes depending on the quantum number and showcases interesting properties regarding its probability density. For quantum numbers greater than 1, there are places in which you are unable to find the particle within the boundary of the well, despite there being 0 potential energy!		Here, we can visualize the wave function and its associated probability density for various quantum levels. As you can see, the wave function changes depending on the quantum number and showcases interesting properties regarding its probability density. For quantum numbers greater than 1, there are places in which you are unable to find the particle within the boundary of the well, despite there being 0 potential energy!

	[[File:Infinite potential.gif]]		[[File:Infinite potential.gif]]

	===Semi-Infinite Potential (Mathematical Model)===		===Semi-Infinite Potential (Mathematical Model)===

	[[File:Semi_infinite.png]]		[[File:Semi_infinite.png]]

	Next, let us consider the case where the boundary at <math>x = L</math> is no longer infinite, but a constant value '''V'''. In this case, we can again solve for the wave function with the Schrodinger Equation, except that we will now have a non-zero term for the potential. The solution to the Schrodinger Equation is as follows:		Next, let us consider the case where the boundary at <math>x = L</math> is no longer infinite, but a constant value '''V'''. In this case, we can again solve for the wave function with the Schrodinger Equation, except that we will now have a non-zero term for the potential. The solution to the Schrodinger Equation is as follows:

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	===Semi-Infinite Potential (Computational Mode)===		===Semi-Infinite Potential (Computational Mode)===
	[[File:Semi_potential.gif]]		[[File:Semi_potential.gif]]

			Here we demonstrate the exponential behavior of the wave function as it enters <math>x \geq L </math>. As seen, the wave function can exist beyond the barrier due to quantum tunneling.
	==Examples==		==Examples==

Eathan at 05:51, 23 April 2022

2022-04-23T05:51:35Z

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	https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/infwell1d/infwell1d.html		https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/infwell1d/infwell1d.html

			https://jakevdp.github.io/blog/2012/09/05/quantum-python/

	[[Category:Which Category did you place this in?]]		[[Category:Which Category did you place this in?]]

Eathan at 05:50, 23 April 2022

2022-04-23T05:50:44Z

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	Classically, if a particle encounters a barrier and does not have sufficient energy to overcome the potential, it will not be allowed to pass through the barrier. However, in this quantum system, the particle can be found outside of the well! As you can see from the piecewise function above, the associated wave function is a decaying exponential. This is the basis of quantum tunneling, whereby a particle can pass through a forbidden region.		Classically, if a particle encounters a barrier and does not have sufficient energy to overcome the potential, it will not be allowed to pass through the barrier. However, in this quantum system, the particle can be found outside of the well! As you can see from the piecewise function above, the associated wave function is a decaying exponential. This is the basis of quantum tunneling, whereby a particle can pass through a forbidden region.

	The distance a particle can penetrate is given by the following expression: <math>\delta x = \frac{h}{\sqrt{2m(V-E}}</math>		The distance a particle can penetrate is given by the following expression: <math>\delta x = \frac{h}{\sqrt{2m(V-E)}}</math>. As you can see, the penetration depth is proportional to Planck's constant, meaning that this distance will be very small.

			===Semi-Infinite Potential (Computational Mode)===
			[[File:Semi_potential.gif]]
	==Examples==		==Examples==

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	==References==		==References==

	~~This section contains the the references you used while writing this page~~		https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_1-Dimensional_box

			Prof. Trebino's Schrodinger Equation Video

			https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/infwell1d/infwell1d.html



	[[Category:Which Category did you place this in?]]		[[Category:Which Category did you place this in?]]

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	== See also ==		== See also ==

	~~Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?~~		===External links===

	~~===Further reading===~~		https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger#Quantum_mechanics

	~~Books, Articles or other print media on this topic~~		https://www.youtube.com/watch?v=2WPA1L9uJqo

	~~===External links===~~		https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_1-Dimensional_box

			https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/infwell1d/infwell1d.html

	~~Internet resources on this topic~~		https://jakevdp.github.io/blog/2012/09/05/quantum-python/

	==References==		==References==