# Worksheet 4A: Introduction to the Quantum Harmonic Oscillator

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*Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.*

## The Classical Harmonic Oscillator

In classical mechanics, a harmonic oscillator (HO) is a system that, when displaced from its equilibrium position (\(x_o\)), experiences a restoring force, \(F\), proportional to the displacement from equilibrium (\(x-x_o\)). For the 1-D HO, this force is given by

\[F(x)=-k(x -x_o) \label{eq1}\]

where \(k\) is a positive constant (often called a "spring constant"). Equation \ref{eq1} is often referred to as Hookes' Law with \(k\) called "Hooke's constant." The force in Equation \ref{eq1} is a conservative force since it dependents only on position. Hence, a potential energy \(V(x)\) can be constructed to describe this force

\[F(x) = \dfrac{- d V(x)}{dx} \label{eq2}\]

Equating Equations \ref{eq1} and \ref{eq2} and then integrating results in the following HO potential

\[V(x) = \dfrac{1}{2} k (x-x_o)^2 \label{eq3}\]

If \(F\) is the only force acting on the system, the system is called a simple (or undamped) harmonic oscillator, and undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). This is demonstrated by solving the coresponding equations of motion (below).

Combining Newton's 2nd law of motion and Equation \ref{eq1} results in a 1D wave equation

\[{F=ma=m{\dfrac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=m{\ddot {x}}=-kx.} \label{eq4}\]

Solving this 2nd order differential equation, we find that the motion is described by the function

\[x(t)=A\cos(\omega t+\varphi ) \label{eq5}\]

where

\[\omega =\sqrt {\dfrac {k}{m}}=\dfrac {2\pi }{T}. \label{eq6}\]

The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, \(A\) with a period \(T\). This motion can be viewed as a particle moving along the HO potential \(V(x)\) or alternatively, as motion of a mass on a simple spring where \(k\) is a measure of the *strength *of the spring.

*A mass connected to a spring that follows Hooke's law is a system that, when displaced from its equilibrium position, experiences a restoring force, \(F\) that is proportional to the displacement, \(x-x_o\).*

### Q1

In the following plots, draw the different classical harmonic oscillator potentials with the given parameters (label the axes too).

- \(V_1(x)\): \(x_o=0\) and \(k=0.5\)
- \(V_2(x)\): \(x_o=2\) and \(k=0.25\)

### Q2

- What is the force exerted on a particle in each of the potentials in Q1 at \(x=-1\), \(x=0\) and \(x=+1\).
- What will happen to this particle if it has no initial motion at these position?

### Q3

For a particle moving in the classical HO potential, the motion will be oscillatory (Equation \ref{eq5}).

- How does the frequency of this oscillation change if the mass of the particle doubles?
- How does the frequency of this oscillation change if the spring constant doubles?

## The Quantum Harmonic Oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics and is one of the few quantum-mechanical systems for which an exact, analytical solution is known (as is the particle in a box discussed previously).

### Q4

We can extend the classical HO to the quantum world by adopting the classical HO potential (Equation \ref{eq3}) into the Schrödinger equation. Write the corresponding time independent Schrödinger equation.

## Solutions to the Quantum Harmonic Oscillator

A particle in a HO potential is trapped just like a particle in a box and similar intuition applies to both systems. While the Schrödinger equation in Q4 can be analytically solved, it is appreciably harder than solving the particle in a box model and is beyond the scope of most introductory quantum classes.

When solved, the eigenstates are

\[ \psi_v(x)= \dfrac {1}{\sqrt {2^v \,v!}} \cdot \left( \dfrac {m\omega }{\pi \hbar }\right)^{1/4} \cdot \exp \left(-\dfrac {m\omega x^2}{2\hbar }\right) \cdot H_{n} \left( \sqrt {\dfrac {m\omega }{\hbar }}x \right), \label{HO1}\]

with \(v=0,1,2,\ldots\).

where the functions \(H_n\) are the *Hermite polynomials.*

\[H_{v}(x)=(-1)^{v}~e^{x^{2}}{\frac {d^{v}}{dx^{v}}}\left(e^{-x^{2}}\right).\]

The corresponding energy levels are

\[ E_{v}=\hbar \omega \left(v+{1 \over 2}\right)\]

### Q5

The quantum HO eigenstates can be decomposed into a three parts:

- Identify the Gaussian component
- Identify the Hermite polynomial component
- Identify the normalization component

### Q6

- What is the zero point energy for the quantum HO? Does it surprise you that it is not zero?
- How many eigenstates does the quantum HO have?
- What are the limits of the quantum HO quantum number and how does it differ from the limits of the quantum number for the particle in a box?

### Q7

What are the boundary conditions (better thought of as expectations for the solutions) for solving the quantum HO?

### Q8

The first two Hermite polynomials are \(H_0=1\) and \(H_1=2x\). From this information and Equation \ref{HO1}, draw the ground-state and first excited-state eigenstates for the quatum HO. Draw the corresponding potential used to generate these eigenstates.

### Q9

- How many nodes are in the two eigenstates drawn in Q8?
- From inspection of your plots, what is the average position (\(\langle x \rangle\)) for the particle in both eigenstates?

### Q10

For a box of similar length (guesstimate it), draw the ground eigenstate and first excited eigenstate for the particle in a box over the corresponding boundary drawn in Q8. How do they differ and how do they resemble each other?