Worksheet 4A Solutions

 

The Classical Harmonic Oscillator

Q1

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

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

Q2

1. What is the force exerted on a particle in each of the potentials in Q1 at \(x= -1\), .\(x= 0\), \(x= \pm 1\)

\(F = -k(x-x_o) \) so:

First Potential:

\(x= -1\)    \(F = -0.5(1-0) \)    \(F = -0.5 \)

\(x= 0\)    \(F = -0.5(0-0) \)    \(F = 0 \)

\(x= \pm 1\)    \(F = -0.5(\pm 1-0) \)    \(F = -0.5 \) and \(F = 0.5 \)

Second Potential:

\(x= -1\)    \(F = -0.25(1-2) \)    \(F = 0.25 \)

\(x= 0\)    \(F = -0.25(0-2) \)    \(F = 0.5 \)

\(x= \pm 1\)    \(F = -0.25(\pm 1-2) \)    \(F = 0.25 \) and \(F = 0.75 \)

 

1. What will happen to this particle if it has no initial motion at these position?

The particle will accelerate to the minimum at all positions unless it is at the minimum. At the minimum, it would remain at rest.

 

Q3

For a particle moving in the classical HO potential, the motion will be oscillatory (Equation Worksheet 4A.5Worksheet 4A.5).

1. How does the frequency of this oscillation change if the mass of the particle doubles?

\( \omega = \sqrt{\frac{k}{m}} \) and \( \omega = 2 \pi \nu \), so as mass increases, the frequency of oscillation decreases.

1. How does the frequency of this oscillation change if the spring constant doubles?

\( \omega = \sqrt{\frac{k}{m}} \) and \( \omega = 2 \pi \nu \), so as \( k \) increases, \( \omega \) increases and frequency increases.

The Quantum Harmonic Oscillator

 

Q4

Write the corresponding time independent Schrödinger equation.

\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{2m}{\hbar^2} (E - \frac{1}{2}k(x-x_o)^2)\psi = 0  \]

Solutions to the Quantum Harmonic Oscillator

 

Q5

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

1. Identify the Gaussian component

\[ \exp \left(-\dfrac {m\omega x^2}{2\hbar }\right) \]

 2. Identify the Hermite polynomial component

\[  H_{n} \left( \sqrt {\dfrac {m \omega }{\hbar }}x \right  \]

3. Identify the normalization component

\[ \dfrac {1}{\sqrt {2^v \,v!}}  \]

Q6

1. What is the zero point energy for the quantum HO? Does it surprise you that it is not zero?

• \[ E_o = \frac{\hbar \omega}{2} \]

2. How many eigenstates does the quantum HO have?

• This is a strange question. There should be an infinite number of eigenstates but at some point, we reach the classical approximation through the correspondence principle.

3. 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?

• The quantum HO starts at \( \nu = 0 \) while the particle in a box starts at \( n = 1 \). 

Q7

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

• The wave function solutions must be symmetric about \( x=0 \), and to be normalizable. The probability density \( | \psi(x) |^2 \) must be finite when integrated from \( -\infty \) to \( \infty \) .
• Eignevalues must be quantized.

Q8

The first two Hermite polynomials are \( H_0=1 \) and \( H_1=2x \). From this information and Equation Worksheet 4A.7, draw the ground-state and first excited-state eigenstates for the quantum HO. Draw the corresponding potential used to generate these eigenstates.

 

Q9

1. How many nodes are in the two eigenstates drawn in Q8?

• 0 and 1

 2. From inspection of your plots, what is the average position (⟨x⟩⟨x⟩) for the particle in both eigenstates?

• 0

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?