Quantum Harmonic Oscillator - Revision history

Ecarder: /* Proving the Ground-State Energy Relation Using Uncertainty Principle */

2022-12-16T06:38:13Z

Proving the Ground-State Energy Relation Using Uncertainty Principle

← Older revision		Revision as of 02:38, 16 December 2022
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	<math>E=\frac{\hbar^2}{8m(\Delta x)^2} + \frac{1}{2}m\omega^2 (\Delta x)^2 </math><br><br>		<math>E=\frac{\hbar^2}{8m(\Delta x)^2} + \frac{1}{2}m\omega^2 (\Delta x)^2 </math><br><br>
	<math>E_0=\frac{\hbar^2}{8m(\sqrt{\frac{\hbar}{2m\omega}})^2} + \frac{1}{2}m\omega^2 (\sqrt{\frac{\hbar}{2m\omega}})^2 </math><br><br>		<math>E_0=\frac{\hbar^2}{8m(\sqrt{\frac{\hbar}{2m\omega}})^2} + \frac{1}{2}m\omega^2 (\sqrt{\frac{\hbar}{2m\omega}})^2 </math><br><br>
	<math>E_0=\frac{2m\omega \hbar}{8m\hbar} + \frac{1}{2}m\omega^2 (\sqrt{\frac{\hbar}{2m\omega}})</math><br><br>		<math>E_0=\frac{2m\omega \hbar}{8m\hbar} + \frac{1}{2}m\omega^2 (\sqrt{\frac{\hbar}{2m\omega}})^2</math><br><br>
	<math>E_0=\frac{\hbar \omega}{4} + \frac{\hbar \omega}{4} </math><br><br>		<math>E_0=\frac{\hbar \omega}{4} + \frac{\hbar \omega}{4} </math><br><br>
	<math> \therefore E_0 = \frac{\hbar \omega}{2} </math> <br>		<math> \therefore E_0 = \frac{\hbar \omega}{2} </math> <br>

Ecarder: /* Proving the Ground-State Energy Relation Using Uncertainty Principle */

2022-12-16T06:36:00Z

Proving the Ground-State Energy Relation Using Uncertainty Principle

← Older revision		Revision as of 02:36, 16 December 2022
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	<math> E=\frac{(\Delta p)^2}{2m} + \frac{1}{2}m\omega^2 (\Delta x)^2 </math><br><br>		<math> E=\frac{(\Delta p)^2}{2m} + \frac{1}{2}m\omega^2 (\Delta x)^2 </math><br><br>
	In considering the lower bound of the Uncertainty Principle, namely when the relationship is equal such that:<br>		In considering the lower bound of the Uncertainty Principle, namely when the relationship is equal such that:<br>
	<math> \Delta x \Delta P = \frac{\hbar}{2} </math> <br><br>		<math> \Delta x \Delta p = \frac{\hbar}{2} </math> <br><br>
	We can rewrite one uncertainty as a function of the other. In this case, we will rewrite the momentum uncertainty as a function of the position uncertainty:<br><br>		We can rewrite one uncertainty as a function of the other. In this case, we will rewrite the momentum uncertainty as a function of the position uncertainty:<br><br>
	<math>\Delta P = \frac{\hbar}{2\Delta x} </math> and substituting into the energy relation <br><br>		<math>\Delta p = \frac{\hbar}{2\Delta x} </math> and substituting into the energy relation <br><br>
	<math> E=\frac{\hbar^2}{8m(\Delta x)^2} + \frac{1}{2}m\omega^2 (\Delta x)^2 </math><br><br>		<math> E=\frac{\hbar^2}{8m(\Delta x)^2} + \frac{1}{2}m\omega^2 (\Delta x)^2 </math><br><br>
	Our goal is to minimize the energy, hence we now differentiate and set it to zero to find the critical point. <br>		Our goal is to minimize the energy, hence we now differentiate and set it to zero to find the critical point. <br>

Ecarder: /* Proving the Ground-State Energy Relation Using Uncertainty Principle */

2022-12-16T06:35:29Z

Proving the Ground-State Energy Relation Using Uncertainty Principle

← Older revision		Revision as of 02:35, 16 December 2022
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	We also know the angular frequency is: <math>\omega=\sqrt{\frac{k}{m}}</math><br><br>		We also know the angular frequency is: <math>\omega=\sqrt{\frac{k}{m}}</math><br><br>
	In solving for <math>k</math>, we get <math>k=\omega^2m</math><br><br>		In solving for <math>k</math>, we get <math>k=\omega^2m</math><br><br>
	~~Rewriting~~ the potential as:<br><br>		Rewrite the potential as:<br><br>
	<math>U(x)=\frac{1}{2}mx^2\omega^2</math><br><br>		<math>U(x)=\frac{1}{2}mx^2\omega^2</math><br><br>
	In formulating in terms of uncertainties, the energy for the QHO must be at least:<br><br>		In formulating in terms of uncertainties, the energy for the QHO must be at least:<br><br>

Ecarder at 01:56, 15 December 2022

2022-12-15T01:56:42Z

@@ Line 15: / Line 15: @@
 This is logical as <math>E_0=0</math> is contradictory to Heisenberg's Uncertainty Principle. If the ground state energy were to be zero, the given particle would be motionless. If the particle is motionless, then <br><br><math>p=0</math> and <math> x=C </math>. Since this hypothetical particle has invariant momentum and position, then <math>\Delta x=0</math> and <math>\Delta p=0</math> which directly violates <math>\Delta p \Delta x \geq \frac{\hbar}{2}</math><br>
-==Deriving the Energy and Considering the Potential for Analytical, Closed-Form Solutions==
+==Proving the Ground-State Energy Relation Using Uncertainty Principle==
-The quantum harmonic oscillator is one of the few systems modeled by the Schrodinger Equation that has an analytical solution. The solution requires many unintuitive substitutions that would be difficult to spontaneously arrive at. We will make an attempt to derive the formula for energy in one spacial dimension using spectral analysis:<br><br>
+The Uncertainty Principle mathematically limits the ground-state energy level to <math> E_0=\frac{\hbar\omega}{2}</math>.<br>
-Classically, we know the spring potential for the SHO: <math>U(x)=\frac{1}{2}kx^2</math><br><br>
+The classical definition of kinetic energy is: <math> KE=\frac{1}{2}mv^2 </math><br>
 We also know the angular frequency is: <math>\omega=\sqrt{\frac{k}{m}}</math><br><br>
 In solving for <math>k</math>, we get <math>k=\omega^2m</math><br><br>
 Rewriting the potential as:<br><br>
 <math>U(x)=\frac{1}{2}mx^2\omega^2</math><br><br>
 The one-dimensional, time-dependent Schrodinger Equation is:<br><br>
@@ Line 85: / Line 112: @@
 Libretexts. (2020, August 11). 5.6: The harmonic-oscillator wavefunctions involve hermite polynomials. Chemistry LibreTexts. Retrieved 2022, from https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/05%3A_The_Harmonic_Oscillator_and_the_Rigid_Rotor/5.06%3A_The_Harmonic-Oscillator_Wavefunctions_involve_Hermite_Polynomials <br> <br>
 Libretexts. (2022, September 12). 7.6: The Quantum Harmonic Oscillator. Physics LibreTexts. Retrieved 2022, from https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator <br><br>
-Born, Max; Jordan, Pascual, Zur Quantenmechanik, Z. f. Physik 34, 858-888 (1925). ZBL51.0728.08. (27 September 1925)
+Born, Max; Jordan, Pascual, Zur Quantenmechanik, Z. f. Physik 34, 858-888 (1925). ZBL51.0728.08. (27 September 1925) <br><br>

Ecarder: /* Connectedness */

2022-12-15T00:43:08Z

Connectedness

← Older revision		Revision as of 20:43, 14 December 2022
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	==Connectedness==		==Connectedness==
	At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.		At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the phenomena that are byproducts of these vibrations.

	==References==		==References==
	Dudik, R. (2004, May 5). The Quantum Harmonic Oscillator. Retrieved 2022, from http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/harmonic/ <br>		Dudik, R. (2004, May 5). The Quantum Harmonic Oscillator. Retrieved 2022, from http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/harmonic/ <br>

Ecarder: /* History */

2022-12-15T00:42:29Z

History

← Older revision		Revision as of 20:42, 14 December 2022
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	<math>H_n(x)=(-1)^ne^{x^2}\cdot \frac{d^n }{d x^n}e^{-x^2}</math><br><br>		<math>H_n(x)=(-1)^ne^{x^2}\cdot \frac{d^n }{d x^n}e^{-x^2}</math><br><br>
	==History==		==History==
	In the case of the quantum harmonic oscillator, there isn't much history for it. It is more so a specific case of the Schrodinger Equation, as it was one of the first cases solved after the famous equation was formulated. The first solution of the QHO I've been able to find was as early as 1925, in a paper written by Max Born and Pascual, Jordan.		In the case of the quantum harmonic oscillator, there isn't much history for it. It is more so a specific case of the Schrodinger Equation, as it was one of the first cases solved after the famous equation was formulated. The first solution of the QHO I've been able to find was as early as 1925, in a paper written by Max Born and Pascual Jordan.

	==Connectedness==		==Connectedness==
	At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.		At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.

Ecarder: /* Comparing the Classic and Quantum */

2022-12-14T16:09:11Z

Comparing the Classic and Quantum

← Older revision		Revision as of 12:09, 14 December 2022
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	then by substitution, we see that <br><br>		then by substitution, we see that <br><br>
	<math>E_0=\frac{\hbar\omega}{2}</math><br><br>		<math>E_0=\frac{\hbar\omega}{2}</math><br><br>
	This is ~~intuitively~~ logical as <math>E_0=0</math> is contradictory to Heisenberg's Uncertainty Principle. If the ground state energy were to be zero, the given particle would be motionless. If the particle is motionless, then <br><br><math>p=0</math> and <math> x=C </math>. Since this hypothetical particle has invariant momentum and position, then <math>\Delta x=0</math> and <math>\Delta p=0</math> which directly violates <math>\Delta p \Delta x \geq \frac{\hbar}{2}</math><br>		This is logical as <math>E_0=0</math> is contradictory to Heisenberg's Uncertainty Principle. If the ground state energy were to be zero, the given particle would be motionless. If the particle is motionless, then <br><br><math>p=0</math> and <math> x=C </math>. Since this hypothetical particle has invariant momentum and position, then <math>\Delta x=0</math> and <math>\Delta p=0</math> which directly violates <math>\Delta p \Delta x \geq \frac{\hbar}{2}</math><br>

	==Deriving the Energy and Considering the Potential for Analytical, Closed-Form Solutions==		==Deriving the Energy and Considering the Potential for Analytical, Closed-Form Solutions==

Ecarder: /* References */

2022-12-14T16:08:13Z

References

← Older revision		Revision as of 12:08, 14 December 2022
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	Dudik, R. (2004, May 5). The Quantum Harmonic Oscillator. Retrieved 2022, from http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/harmonic/ <br>		Dudik, R. (2004, May 5). The Quantum Harmonic Oscillator. Retrieved 2022, from http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/harmonic/ <br>

	Libretexts. (2020, August 11). 5.6: The harmonic-oscillator wavefunctions involve hermite polynomials. Chemistry LibreTexts. Retrieved 2022, from https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/05%3A_The_Harmonic_Oscillator_and_the_Rigid_Rotor/5.06%3A_The_Harmonic-Oscillator_Wavefunctions_involve_Hermite_Polynomials <br>		Libretexts. (2020, August 11). 5.6: The harmonic-oscillator wavefunctions involve hermite polynomials. Chemistry LibreTexts. Retrieved 2022, from https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/05%3A_The_Harmonic_Oscillator_and_the_Rigid_Rotor/5.06%3A_The_Harmonic-Oscillator_Wavefunctions_involve_Hermite_Polynomials <br> <br>
	Libretexts. (2022, September 12). 7.6: The Quantum Harmonic Oscillator. Physics LibreTexts. Retrieved 2022, from https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator <br>		Libretexts. (2022, September 12). 7.6: The Quantum Harmonic Oscillator. Physics LibreTexts. Retrieved 2022, from https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator <br><br>
	Born, Max; Jordan, Pascual, Zur Quantenmechanik, Z. f. Physik 34, 858-888 (1925). ZBL51.0728.08. (27 September 1925)		Born, Max; Jordan, Pascual, Zur Quantenmechanik, Z. f. Physik 34, 858-888 (1925). ZBL51.0728.08. (27 September 1925)

Ecarder at 16:07, 14 December 2022

2022-12-14T16:07:35Z

← Older revision		Revision as of 12:07, 14 December 2022
Line 1:		Line 1:
	'''Claimed - Eric Carder Fall 2022''' <br>		'''Claimed - Eric Carder Fall 2022''' <br>
	The Quantum Harmonic Oscillator is the quantum parallel to the classical simple harmonic oscillator. The key difference between these two is in the name. In the quantum harmonic oscillator, energy levels are ''quantized'' meaning there are discrete energy levels to this oscillator, (it cannot be any positive value as a classical oscillator can have). At low levels of energy, an oscillator obeys the rules of quantum mechanics. ~~So the~~ QHO ~~has numerous~~ applications ~~such~~ as molecular vibrations. These energy levels, denoted by <math>E_n, n=1,2,3... </math> ~~and is~~ evaluated by the relation: <br><math> E_n=(n+\frac{1}{2})\hbar\omega </math><br>		The Quantum Harmonic Oscillator is the quantum parallel to the classical simple harmonic oscillator. The key difference between these two is in the name. In the quantum harmonic oscillator, energy levels are ''quantized'' meaning there are discrete energy levels to this oscillator, (it cannot be any positive value as a classical oscillator can have). At low levels of energy, an oscillator obeys the rules of quantum mechanics. The QHO does have applications, most notably as an approximation of molecular vibrations. These energy levels, denoted by <math>E_n, n=1,2,3... </math> can be evaluated by the relation: <br><math> E_n=(n+\frac{1}{2})\hbar\omega </math><br>
	Where <math>n</math> is the principal quantum number, <math>\hbar</math> is the reduced planks constant, and <math>\omega</math> is the angular frequency of the oscillator.<br>		Where <math>n</math> is the principal quantum number, <math>\hbar</math> is the reduced planks constant, and <math>\omega</math> is the angular frequency of the oscillator.<br>
	[[File:qho2.png\|300px\|]]<br>		[[File:qho2.png\|300px\|]]<br>
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	Where <math> H_n </math> are Hermite Polynomials defined as: <br><br>		Where <math> H_n </math> are Hermite Polynomials defined as: <br><br>
	<math>H_n(x)=(-1)^ne^{x^2}\cdot \frac{d^n }{d x^n}e^{-x^2}</math><br><br>		<math>H_n(x)=(-1)^ne^{x^2}\cdot \frac{d^n }{d x^n}e^{-x^2}</math><br><br>
			==History==
			In the case of the quantum harmonic oscillator, there isn't much history for it. It is more so a specific case of the Schrodinger Equation, as it was one of the first cases solved after the famous equation was formulated. The first solution of the QHO I've been able to find was as early as 1925, in a paper written by Max Born and Pascual, Jordan.
	==Connectedness==		==Connectedness==
	At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.		At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.
Line 81:		Line 82:

	Libretexts. (2020, August 11). 5.6: The harmonic-oscillator wavefunctions involve hermite polynomials. Chemistry LibreTexts. Retrieved 2022, from https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/05%3A_The_Harmonic_Oscillator_and_the_Rigid_Rotor/5.06%3A_The_Harmonic-Oscillator_Wavefunctions_involve_Hermite_Polynomials <br>		Libretexts. (2020, August 11). 5.6: The harmonic-oscillator wavefunctions involve hermite polynomials. Chemistry LibreTexts. Retrieved 2022, from https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/05%3A_The_Harmonic_Oscillator_and_the_Rigid_Rotor/5.06%3A_The_Harmonic-Oscillator_Wavefunctions_involve_Hermite_Polynomials <br>
	Libretexts. (2022, September 12). 7.6: The Quantum Harmonic Oscillator. Physics LibreTexts. Retrieved 2022, from https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator		Libretexts. (2022, September 12). 7.6: The Quantum Harmonic Oscillator. Physics LibreTexts. Retrieved 2022, from https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator <br>
			Born, Max; Jordan, Pascual, Zur Quantenmechanik, Z. f. Physik 34, 858-888 (1925). ZBL51.0728.08. (27 September 1925)

Ecarder: /* Deriving the Energy and Potential for Analytical, Solutions */

2022-12-07T17:59:05Z

Deriving the Energy and Potential for Analytical, Solutions

← Older revision		Revision as of 13:59, 7 December 2022
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	This is intuitively logical as <math>E_0=0</math> is contradictory to Heisenberg's Uncertainty Principle. If the ground state energy were to be zero, the given particle would be motionless. If the particle is motionless, then <br><br><math>p=0</math> and <math> x=C </math>. Since this hypothetical particle has invariant momentum and position, then <math>\Delta x=0</math> and <math>\Delta p=0</math> which directly violates <math>\Delta p \Delta x \geq \frac{\hbar}{2}</math><br>		This is intuitively logical as <math>E_0=0</math> is contradictory to Heisenberg's Uncertainty Principle. If the ground state energy were to be zero, the given particle would be motionless. If the particle is motionless, then <br><br><math>p=0</math> and <math> x=C </math>. Since this hypothetical particle has invariant momentum and position, then <math>\Delta x=0</math> and <math>\Delta p=0</math> which directly violates <math>\Delta p \Delta x \geq \frac{\hbar}{2}</math><br>

	==Deriving the Energy and Potential for Analytical, Solutions==		==Deriving the Energy and Considering the Potential for Analytical, Closed-Form Solutions==
	The quantum harmonic oscillator is one of the few systems modeled by the Schrodinger Equation that has an analytical solution. The solution requires many unintuitive substitutions that would be difficult to spontaneously arrive at. We will make an attempt to derive the formula for energy in one spacial dimension using spectral analysis:<br><br>		The quantum harmonic oscillator is one of the few systems modeled by the Schrodinger Equation that has an analytical solution. The solution requires many unintuitive substitutions that would be difficult to spontaneously arrive at. We will make an attempt to derive the formula for energy in one spacial dimension using spectral analysis:<br><br>
	Classically, we know the spring potential for the SHO: <math>U(x)=\frac{1}{2}kx^2</math><br><br>		Classically, we know the spring potential for the SHO: <math>U(x)=\frac{1}{2}kx^2</math><br><br>
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	<math>\frac{d\Psi(t)}{dt}=\frac{du(t)}{dt}e^{\frac{-t^2}{2}}-tu(t)e^{\frac{-t^2}{2}}</math><br><br>		<math>\frac{d\Psi(t)}{dt}=\frac{du(t)}{dt}e^{\frac{-t^2}{2}}-tu(t)e^{\frac{-t^2}{2}}</math><br><br>

	<math>\frac{d^2\Psi(t)}{dt^2}e^{\frac{-t^2}{2}}-tu(t)e^{\frac{-t^2}{2}}-tu(t)e^{\frac{-t^2}{2}}+u(t)(t^2-1)e^{\frac{-t^2}{2}}.</math><br><br>		<math>\frac{d^2\Psi(t)}{dt^2}=\frac{d^2u(t)}{dt^2}e^{\frac{-t^2}{2}}-t\frac{du(t)}{dt}e^{\frac{-t^2}{2}}-t\frac{du(t)}{dt}e^{\frac{-t^2}{2}}+u(t)(t^2-1)e^{\frac{-t^2}{2}}.</math><br><br>

	Time to substitute the derivatives:		Time to substitute the derivatives:
	<math>\frac{d^~~2\Psi~~(t)}{dt^2}e^{\frac{-t^2}{2}}-2t\frac{du(t)}{dt}+u(t)(t^2-1)e^{\frac{-t^2}{2}}+(\frac{2E}{\hbar\omega}-t^2)u(t)e^{\frac{-t^2}{2}}=0</math><br><br>		<math>\frac{d^2u(t)}{dt^2}e^{\frac{-t^2}{2}}-2t\frac{du(t)}{dt}+u(t)(t^2-1)e^{\frac{-t^2}{2}}+(\frac{2E}{\hbar\omega}-t^2)u(t)e^{\frac{-t^2}{2}}=0</math><br><br>
	Which simplifies quite nicely:<br><br>		Which simplifies quite nicely:<br><br>
	<math>\frac{d^~~2\Psi~~(t)}{dt^2}e^{\frac{-t^2}{2}}-2t\frac{du(t)}{dt}-u(t)e^{\frac{-t^2}{2}}+\frac{2E}{\hbar\omega}u(t)e^{\frac{-t^2}{2}}=0</math><br><br>		<math>\frac{d^2u(t)}{dt^2}e^{\frac{-t^2}{2}}-2t\frac{du(t)}{dt}-u(t)e^{\frac{-t^2}{2}}+\frac{2E}{\hbar\omega}u(t)e^{\frac{-t^2}{2}}=0</math><br><br>
	Now we omit <math>e^{\frac{-t^2}{2}}</math> by dividing both sides by the expression:<br><br>		Now we omit <math>e^{\frac{-t^2}{2}}</math> by dividing both sides by the expression:<br><br>
	<math>\frac{d^~~2\Psi~~(t)}{dt^2}-2t\frac{du(t)}{dt}-u(t)+(\frac{2E}{\hbar\omega}-1)u(t)=0</math><br><br>		<math>\frac{d^2u(t)}{dt^2}-2t\frac{du(t)}{dt}-u(t)+(\frac{2E}{\hbar\omega}-1)u(t)=0</math><br><br>

	We can now construct a power series to find a direct solution:<br><br>		We can now construct a power series to find a direct solution:<br><br>
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	Where <math> H_n </math> are Hermite Polynomials defined as: <br><br>		Where <math> H_n </math> are Hermite Polynomials defined as: <br><br>
	<math>H_n(x)=(-1)^ne^{x^2}\cdot \frac{d^n }{d x^n}e^{-x^2}</math><br><br>		<math>H_n(x)=(-1)^ne^{x^2}\cdot \frac{d^n }{d x^n}e^{-x^2}</math><br><br>

	==Connectedness==		==Connectedness==
	At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.		At first glance, applications of the quantum harmonic oscillator seem limited. It's important to realize that it is simply an analog of its classical counterpart. When harmonically oscillating systems approach the atomic scale, they must obey the laws of quantum mechanics as anything else would. An adept understanding of quantum harmonic oscillators can enhance humans' understanding of larger-scale models. A quick example would be the consideration of how an understanding of molecular vibrations can lead to a greater understanding of the byproducts of these vibrations.