Solution for Simple Harmonic Oscillator - Revision history

Lim, Xuen Zhen: Added hyperlink to Morse potential

2022-04-25T03:46:25Z

Added hyperlink to Morse potential

← Older revision		Revision as of 23:46, 24 April 2022
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	===Morse Potential===		===Morse Potential===
	Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:		[//en.wikipedia.org/wiki/Morse_potential Morse potential], named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

	<math>V(x) = A (1-e^{-a(x-x_{0})})^2</math>		<math>V(x) = A (1-e^{-a(x-x_{0})})^2</math>

Lim, Xuen Zhen: Up the hierarchy of every title and subtitle by one.

2022-04-25T03:41:54Z

Up the hierarchy of every title and subtitle by one.

← Older revision		Revision as of 23:41, 24 April 2022
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	<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>		<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>

	===Introduction===		==Introduction==
	One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.		One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

	===Mathematical Setup===		==Mathematical Setup==
	We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.		We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.
	<br>		<br>
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	The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.		The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.

	===Deriving the Solution===		==Deriving the Solution==
	====Finding Ground State Wave Function====		===Finding Ground State Wave Function===
	For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as <math> x→+∞ </math> and <math> x→-∞ </math>. A simple general wave function that satisfies this requirement is <math> \Psi (x) = A e^{-ax^2} </math>. We begin the derivation with finding the second order differential of the general wave equation.		For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as <math> x→+∞ </math> and <math> x→-∞ </math>. A simple general wave function that satisfies this requirement is <math> \Psi (x) = A e^{-ax^2} </math>. We begin the derivation with finding the second order differential of the general wave equation.
	<br><br>		<br><br>
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	One interesting trait of this wave function is, just like the finite potential well, the probability density can pentrate into the forbidden region beyond the classical turning points (the boundary of the region at which the potential energy becomes higher than the energy).		One interesting trait of this wave function is, just like the finite potential well, the probability density can pentrate into the forbidden region beyond the classical turning points (the boundary of the region at which the potential energy becomes higher than the energy).

	====Finding energy function====		===Finding energy function===

	With <math>a</math> derived above, we can easily substitute it into the second equation and find the energy <math>E</math>.		With <math>a</math> derived above, we can easily substitute it into the second equation and find the energy <math>E</math>.
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	Note that, contrary to one-dimensional energy function of a potential well, the energy function here are uniformly spaced with the interval <math>∆E=\hbar w_{0}</math>.		Note that, contrary to one-dimensional energy function of a potential well, the energy function here are uniformly spaced with the interval <math>∆E=\hbar w_{0}</math>.

	===Applications===		==Applications==
	While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule like HCl. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.		While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule like HCl. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.

	====Morse Potential====		===Morse Potential===
	Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:		Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

Lim, Xuen Zhen: Added even more content to morse potential.

2022-04-25T03:40:31Z

Added even more content to morse potential.

← Older revision		Revision as of 23:40, 24 April 2022
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	Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:		Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

	<math>V(x) = ~~D_{e}~~ (1-e^{-a(x-x_{0})})^2</math>		<math>V(x) = A (1-e^{-a(x-x_{0})})^2</math>
	<ul>		<ul>
	<li><math>~~D_{e}~~</math> is the depth of the well (defined relative to the dissociated atoms)</li>		<li><math>A</math> is the depth of the well (defined relative to the dissociated atoms)</li>
	<li><math>x</math> is the distance between atoms</li>		<li><math>x</math> is the distance between atoms</li>
	<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>		<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>
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	<br>		<br>

	We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions by		We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions as seen below.

	<math>V_{morse} (x)≈ \frac{1}{2}</math> <math>2 a^2 A</math> <math>x^2</math>		<math>V_{morse} (x)≈ \frac{1}{2}</math> <span style="color:red"><math>2 a^2 A</math></span> <math>x^2</math>
	<math>V_{harmonic} (x)= \frac{1}{2}</math> <math>k</math> <math>x^2</math>		<math>V_{harmonic} (x)= \frac{1}{2}</math> <span style="color:red"><math>k</math></span> <math>x^2</math>
	<br>		<br>

	which shows that we may approximate the Schrodinger equation by treating <math>k</math> <math>2 a^2 A</math> since these are both constants.		which shows that we may approximate the Schrodinger equation by treating <math>k</math> as <math>2 a^2 A</math> since these are both constants. We can find the solution for ground state wave function and energy level function by simply replacing the <math>k</math> in the solutions for harmonic oscillator with <math>2 a^2 A</math>. Note that the <math>m</math> used for Morse potential related derivations must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math> as we are working with a two-particle system rather than a singular one.

			Finding the approximate energy level function of Morse potential at low energy levels:

			<math> E_{n} = (n+\frac{1}{2}) \hbar \sqrt{\frac{k}{m}} </math>
			<br>
			<math> E_{n} = (n+\frac{1}{2}) \hbar a \sqrt{\frac{2 A}{m}} </math>

			Note that the <math>m</math> used here must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math>.

			Applying similar logic to approximate the ground state wave function of Morse potential:

			<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
			<br>
			<math >\Psi (x) = (\frac{m \sqrt{k/m}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
			<br>
			<math >\Psi (x) = (\frac{m a\sqrt{2 A/m}}{\hbar \pi})^{1/4} e^{-(a\sqrt{2 A m}/2\hbar) x^2}</math>
			<br>

Lim, Xuen Zhen: Added content to Morse potential application

2022-04-25T03:21:58Z

Added content to Morse potential application

← Older revision		Revision as of 23:21, 24 April 2022
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	<math>V(x) = D_{e} (1-e^{-a(x-x_{0})})^2</math>		<math>V(x) = D_{e} (1-e^{-a(x-x_{0})})^2</math>
			<ul>
			<li><math>D_{e}</math> is the depth of the well (defined relative to the dissociated atoms)</li>
			<li><math>x</math> is the distance between atoms</li>
			<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>
			<li><math>a</math> controls the width of the potential (smaller a leads to wider well)</li>
			</ul>

			The Morse potential function, by itself, is a complicated function that would make the Schrodinger equation difficult to solve. However, when overlaying the Morse potential curve on top of the harmonic oscillator potential curve, we can observe that they are similar near the point of equilibrium <math>x_{0}</math>. Due to this, we may relate the Morse potential to harmonic oscillator when working at low energy levels to get a rather accurate approximation of the wave function and energy function conveniently. To achieve this, we need a simplified, approximated function of Morse potential around the point of equilibrium, which can be done by performing [//en.wikipedia.org/wiki/Taylor_series Taylor Series] expansion about <math>x_{0}</math>.

			<math>V(x_{0})+\frac{d}{d x} V(x_{0}) x + \frac{1}{2}\frac{d^2}{d x^2} V(x_{0}) x^2 + ... </math>
			<br>
			<math>= 0 + 0•x + \frac{1}{2} 2 a^2 A x^2 + ...</math>
			<br>
			<math>≈ \frac{1}{2} 2 a^2 A x^2</math>
			<br>

			We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions by

			<math>V_{morse} (x)≈ \frac{1}{2}</math> <math>2 a^2 A</math> <math>x^2</math>
			<math>V_{harmonic} (x)= \frac{1}{2}</math> <math>k</math> <math>x^2</math>
			<br>

			which shows that we may approximate the Schrodinger equation by treating <math>k</math> <math>2 a^2 A</math> since these are both constants.

Lim, Xuen Zhen: /* Morse Potential */

2022-04-25T02:52:29Z

Morse Potential

← Older revision		Revision as of 22:52, 24 April 2022
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	====Morse Potential====		====Morse Potential====
			Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

			<math>V(x) = D_{e} (1-e^{-a(x-x_{0})})^2</math>

Lim, Xuen Zhen: Changed potential symbol from U to V

2022-04-25T02:42:58Z

Changed potential symbol from U to V

← Older revision		Revision as of 22:42, 24 April 2022
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	===Introduction===		===Introduction===
	One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.		One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

	===Mathematical Setup===		===Mathematical Setup===
	We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> U </math> with <math> \frac{1}{2} k x^2 </math>.		We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.
	<br>		<br>
	<math>\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{d x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>		<math>\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{d x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>

Lim, Xuen Zhen: Added hyperlink to Gaussian Integral

2022-04-25T02:38:50Z

Added hyperlink to Gaussian Integral

← Older revision		Revision as of 22:38, 24 April 2022
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	<math> 1 = \int_{-∞}^{+∞} \|\Psi\|^2 dx = \int_{-∞}^{+∞} A^2 e^{-(\sqrt{k m}/\hbar) x^2} dx </math>		<math> 1 = \int_{-∞}^{+∞} \|\Psi\|^2 dx = \int_{-∞}^{+∞} A^2 e^{-(\sqrt{k m}/\hbar) x^2} dx </math>

	Using the Gaussian Integral ~~identity~~ <math> 1 = \int_{-∞}^{+∞} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} </math>, the integral is turned into		Using the [//en.wikipedia.org/wiki/Gaussian_integral Gaussian Integral], <math> 1 = \int_{-∞}^{+∞} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} </math>, the integral is turned into

	<math>1 = A^2 \sqrt{\frac{\hbar \pi}{\sqrt{k m}}} </math>		<math>1 = A^2 \sqrt{\frac{\hbar \pi}{\sqrt{k m}}} </math>

Lim, Xuen Zhen: Corrected hyperlink syntax

2022-04-25T02:35:47Z

Corrected hyperlink syntax

← Older revision		Revision as of 22:35, 24 April 2022
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	===Introduction===		===Introduction===
	One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common ~~<a href="https:~~//www.physicsbook.gatech.edu/Spring_Potential_Energy">classical oscillator: a spring~~</a>~~. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.		One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

	===Mathematical Setup===		===Mathematical Setup===

Lim, Xuen Zhen: Corrected a href syntax error

2022-04-25T02:29:05Z

Corrected a href syntax error

← Older revision		Revision as of 22:29, 24 April 2022
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	===Introduction===		===Introduction===
	One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common <a href="https://www.physicsbook.gatech.edu/Spring_Potential_Energy">classical oscillator: a spring<a>. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.		One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common <a href="https://www.physicsbook.gatech.edu/Spring_Potential_Energy">classical oscillator: a spring</a>. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

	===Mathematical Setup===		===Mathematical Setup===

Lim, Xuen Zhen: Embedded a link to Physics 1 Spring Potential Energy page.

2022-04-25T02:25:41Z

Embedded a link to Physics 1 Spring Potential Energy page.

← Older revision		Revision as of 22:25, 24 April 2022
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	===Introduction===		===Introduction===
	One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common classical oscillator: ~~an object attached to~~ a spring. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.		One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common <a href="https://www.physicsbook.gatech.edu/Spring_Potential_Energy">classical oscillator: a spring<a>. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

	===Mathematical Setup===		===Mathematical Setup===