Chapter 4 Exercises: The Quantum Harmonic Oscillator

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 4 Exercises: The Quantum Harmonic Oscillator

Problems are organized by section and graded by difficulty: - [B] Basic — direct application of formulas and definitions - [I] Intermediate — requires multi-step reasoning or combining concepts - [A] Advanced — requires deeper analysis, proof, or synthesis

Section 4.1–4.2: Classical Context and Universality

Problem 4.1 [B]

The Morse potential for a diatomic molecule is $V(r) = D_e\left(1 - e^{-a(r-r_0)}\right)^2$, where $D_e$ is the dissociation energy, $a$ controls the width, and $r_0$ is the equilibrium bond length.

(a) Expand $V(r)$ to second order around $r = r_0$ and identify the effective spring constant $k_{\text{eff}}$.

(b) For HCl, $D_e = 4.62$ eV and $a = 1.87$ \AA$^{-1}$. Calculate $k_{\text{eff}}$ and the classical oscillation frequency $\omega = \sqrt{k_{\text{eff}}/\mu}$, using the reduced mass $\mu = m_H m_{\text{Cl}}/(m_H + m_{\text{Cl}})$.

(c) At what vibrational quantum number does the harmonic approximation begin to break down significantly? (Hint: compare $(n + \frac{1}{2})\hbar\omega$ with $D_e$.)

Problem 4.2 [B]

A classical harmonic oscillator has mass $m = 0.5$ kg and spring constant $k = 200$ N/m.

(a) Calculate the angular frequency $\omega$.

(b) If the amplitude is $A = 0.1$ m, what is the total energy?

(c) What is the quantum number $n$ for this energy? (Calculate $(n + \frac{1}{2})\hbar\omega = E$ and solve for $n$.)

(d) Comment on the observability of energy quantization for macroscopic oscillators.

Problem 4.3 [I]

Show that the Taylor expansion argument works for any well-behaved potential with a stable minimum, by considering the general quartic potential $V(x) = \alpha x^2 + \beta x^3 + \gamma x^4$ (with $\alpha > 0$). Find the effective harmonic frequency $\omega$ and estimate the energy scale at which the cubic and quartic terms become important relative to the quadratic term.

Section 4.3: Analytical Solution

Problem 4.4 [B]

Verify by direct substitution that $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2 / 2\hbar}$ satisfies the time-independent Schr\u00f6dinger equation with $V = \frac{1}{2}m\omega^2 x^2$ and $E = \frac{1}{2}\hbar\omega$.

Problem 4.5 [B]

(a) Using the recursion relation $a_{j+2} = \frac{2j + 1 - \epsilon}{(j+1)(j+2)} a_j$, compute the first four coefficients $a_0, a_2, a_4, a_6$ of the even series for $\epsilon = 5$ (i.e., $n = 2$). Verify that the series terminates at $a_4$.

(b) Show that the resulting polynomial is proportional to $H_2(\xi) = 4\xi^2 - 2$.

Problem 4.6 [I]

Using the Rodrigues formula $H_n(\xi) = (-1)^n e^{\xi^2} \frac{d^n}{d\xi^n} e^{-\xi^2}$:

(a) Derive $H_0(\xi)$, $H_1(\xi)$, $H_2(\xi)$, and $H_3(\xi)$.

(b) Verify the recursion relation $H_{n+1}(\xi) = 2\xi H_n(\xi) - 2n H_{n-1}(\xi)$ for $n = 1$ and $n = 2$.

(c) Show that $H_n(\xi)$ satisfies Hermite's differential equation $H_n'' - 2\xi H_n' + 2nH_n = 0$.

Problem 4.7 [I]

Calculate the normalization constant for $\psi_3(x)$ by evaluating $\int_{-\infty}^{\infty} |H_3(\xi)|^2 e^{-\xi^2} d\xi$ using the orthogonality relation for Hermite polynomials. Verify that your result agrees with the general formula $\frac{1}{\sqrt{2^n n!}}$.

Problem 4.8 [A]

Generating function proof. The generating function for Hermite polynomials is $e^{2\xi t - t^2} = \sum_{n=0}^{\infty} \frac{H_n(\xi)}{n!} t^n$.

(a) Use this generating function to prove the orthogonality relation: $$\int_{-\infty}^{\infty} H_m(\xi) H_n(\xi) e^{-\xi^2} d\xi = \sqrt{\pi}\, 2^n n!\, \delta_{mn}$$

(b) Derive the recursion relation $H_{n+1} = 2\xi H_n - 2nH_{n-1}$ from the generating function.

Problem 4.9 [A]

Show that for large $n$, the classical turning points of the QHO are at $x_{\text{tp}} = \pm\sqrt{(2n+1)\hbar/m\omega}$.

(a) Calculate the ratio of the quantum probability density at the turning point to the maximum probability density at $x = 0$ for the ground state.

(b) For $n = 20$, use the asymptotic approximation of Hermite polynomials to argue that the probability density peaks near the classical turning points.

Section 4.4: Algebraic (Ladder Operator) Method

Problem 4.10 [B]

Express $\hat{x}$ and $\hat{p}$ in terms of $\hat{a}$ and $\hat{a}^\dagger$. Then calculate:

(a) $\langle 3|\hat{x}|4\rangle$

(b) $\langle 5|\hat{p}|4\rangle$

(c) $\langle 2|\hat{x}^2|2\rangle$

Problem 4.11 [B]

Calculate the following, showing all steps:

(a) $\hat{a}^2|5\rangle$

(b) $(\hat{a}^\dagger)^3|2\rangle$

(c) $\hat{a}\hat{a}^\dagger|n\rangle$

(d) $\hat{a}^\dagger\hat{a}\hat{a}^\dagger|n\rangle$

Problem 4.12 [I]

Using the ladder operators, prove the selection rule for the harmonic oscillator: $\langle m|\hat{x}|n\rangle = 0$ unless $m = n \pm 1$. This explains why the QHO only absorbs or emits one quantum of energy at a time in electric dipole transitions.

Problem 4.13 [I]

Compute the matrix representation of $\hat{H}$, $\hat{x}$, and $\hat{p}$ in the $\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\}$ basis. Write them as explicit $4 \times 4$ matrices.

Problem 4.14 [I]

Prove that $[\hat{a}^2, (\hat{a}^\dagger)^2] = 4\hat{a}^\dagger\hat{a} + 2$ using the fundamental commutator $[\hat{a}, \hat{a}^\dagger] = 1$.

Hint: Use the identity $[\hat{A}\hat{B}, \hat{C}] = \hat{A}[\hat{B}, \hat{C}] + [\hat{A}, \hat{C}]\hat{B}$ iteratively.

Problem 4.15 [I]

Show that for any energy eigenstate $|n\rangle$:

(a) $\langle\hat{x}\rangle = 0$ and $\langle\hat{p}\rangle = 0$

(b) $\langle\hat{x}^2\rangle = \frac{\hbar}{2m\omega}(2n + 1)$ and $\langle\hat{p}^2\rangle = \frac{m\omega\hbar}{2}(2n + 1)$

(c) $\Delta x\cdot\Delta p = (n + \frac{1}{2})\hbar$

(d) Verify that the ground state saturates the uncertainty bound.

Problem 4.16 [A]

The number operator $\hat{n} = \hat{a}^\dagger\hat{a}$ counts quanta. Prove:

(a) $[\hat{n}, \hat{a}^k] = -k\hat{a}^k$ for positive integer $k$.

(b) $[\hat{n}, (\hat{a}^\dagger)^k] = k(\hat{a}^\dagger)^k$.

(c) Use part (a) to show that $\hat{a}^k|n\rangle \propto |n-k\rangle$ (for $n \geq k$) and find the proportionality constant.

Problem 4.17 [A]

The virial theorem for the QHO. Show that for any energy eigenstate:

$$\langle T\rangle = \langle V\rangle = \frac{1}{2}E_n$$

where $T = \hat{p}^2/2m$ and $V = \frac{1}{2}m\omega^2\hat{x}^2$. Do this:

(a) Using the explicit expectation values from Problem 4.15.

(b) By operator methods: consider $[\hat{H}, \hat{x}\hat{p}]$ and use the fact that its expectation value in a stationary state vanishes.

Section 4.5: Zero-Point Energy

Problem 4.18 [B]

(a) Calculate the zero-point energy of the CO molecule ($\omega = 4.09 \times 10^{14}$ rad/s) in eV.

(b) Compare this with the thermal energy $k_B T$ at room temperature ($T = 300$ K). Is the zero-point energy significant at room temperature?

(c) At what temperature does $k_B T$ equal $\hbar\omega$ for CO?

Problem 4.19 [I]

Uncertainty principle derivation of zero-point energy.

(a) Write the energy $E = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2$ and substitute $p \sim \Delta p \sim \hbar/(2\Delta x)$ (from the uncertainty principle), $x^2 \sim (\Delta x)^2$.

(b) Minimize the resulting expression with respect to $\Delta x$ and show that the minimum energy is $E_{\min} = \frac{1}{2}\hbar\omega$.

(c) Explain why this argument gives the exact answer (not just an estimate).

Problem 4.20 [I]

Isotope effects. Deuterium (D) has approximately twice the mass of hydrogen (H).

(a) For the same spring constant $k$, how does the zero-point energy of a D-X bond compare to an H-X bond?

(b) The O-H stretching frequency is $\omega_{\text{OH}} \approx 6.9 \times 10^{14}$ rad/s. What is the difference in zero-point energy between O-H and O-D bonds?

(c) Explain qualitatively why D$_2$O has a higher boiling point than H$_2$O (by 1.4 $^\circ$C) in terms of zero-point energy.

Section 4.6: Wavefunctions and Visualization

Problem 4.21 [B]

For the ground state of the QHO, calculate the probability of finding the particle beyond the classical turning point:

$$P_{\text{forbidden}} = 2\int_{x_{\text{tp}}}^{\infty} |\psi_0(x)|^2\, dx$$

Express your answer in terms of the complementary error function $\text{erfc}(z) = \frac{2}{\sqrt{\pi}}\int_z^{\infty} e^{-t^2}dt$, and evaluate numerically. (The answer is approximately 16%.)

Problem 4.22 [I]

Node counting and the oscillation theorem.

(a) Plot (or compute) $|\psi_n(x)|^2$ for $n = 0, 1, 2, 5, 10$. Verify that $\psi_n$ has $n$ nodes.

(b) Show that between any two consecutive nodes of $\psi_n$, there is exactly one node of $\psi_{n+1}$. (This is the Sturm oscillation theorem.)

(c) For $n = 10$, compare the envelope of $|\psi_{10}(x)|^2$ with the classical probability distribution $\rho(x) = \frac{1}{\pi\sqrt{A^2 - x^2}}$ (where $A$ is the classical amplitude for energy $E_{10}$).

Problem 4.23 [I]

Superposition dynamics. Consider the state $|\Psi(0)\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$.

(a) Write $\Psi(x, t)$ explicitly using the QHO wavefunctions.

(b) Calculate $\langle\hat{x}\rangle(t)$. Show that it oscillates at frequency $\omega$.

(c) Calculate $\langle\hat{x}^2\rangle(t)$ and show that the uncertainty in position is time-dependent.

(d) Compare this behavior with a coherent state. In what sense is the coherent state "more classical"?

Section 4.7: Coherent States

Problem 4.24 [B]

For a coherent state $|\alpha\rangle$ with $\alpha = 3$:

(a) Calculate the mean number of quanta $\bar{n}$ and the standard deviation $\Delta n$.

(b) Find the probabilities $P(n)$ for $n = 0, 1, 2, \ldots, 10$ and plot the distribution.

(c) What is the most probable value of $n$? (Hint: it is the integer nearest to $|\alpha|^2 - 1$.)

Problem 4.25 [I]

Proving coherent states are minimum uncertainty.

(a) Calculate $\langle\hat{x}\rangle$, $\langle\hat{x}^2\rangle$, $\langle\hat{p}\rangle$, $\langle\hat{p}^2\rangle$ for the coherent state $|\alpha\rangle$.

(b) Show that $\Delta x \cdot \Delta p = \hbar/2$ for any $\alpha$.

(c) Explain why this does not depend on $\alpha$, even though the mean position and momentum do.

Problem 4.26 [I]

Show that coherent states are not orthogonal: $\langle\beta|\alpha\rangle = e^{-|\alpha|^2/2 - |\beta|^2/2 + \beta^*\alpha} \neq 0$.

Calculate $|\langle\beta|\alpha\rangle|^2$ for $\alpha = 1, \beta = 2$. Despite non-orthogonality, coherent states form an overcomplete set: $\frac{1}{\pi}\int |\alpha\rangle\langle\alpha|\, d^2\alpha = \hat{I}$. This overcomplete resolution of identity is useful in quantum optics.

Problem 4.27 [A]

Time evolution of coherent states.

(a) Show that $e^{-i\hat{H}t/\hbar}|\alpha\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle$.

(b) Calculate $\langle\hat{x}\rangle(t)$ and $\langle\hat{p}\rangle(t)$ for the time-evolved coherent state. Show they satisfy the classical equations of motion.

(c) Show that $|\psi(x,t)|^2$ is a Gaussian of constant width that oscillates without spreading. Why is this special to the harmonic potential?

Problem 4.28 [A]

Squeezed states (preview of Ch 27). A squeezed state has $\Delta x < \sqrt{\hbar/2m\omega}$ at the expense of $\Delta p > \sqrt{m\omega\hbar/2}$ (or vice versa), while still maintaining $\Delta x \cdot \Delta p = \hbar/2$.

(a) If $\Delta x = \frac{1}{r}\sqrt{\hbar/2m\omega}$ for squeeze parameter $r > 1$, what is $\Delta p$?

(b) In which applications would you want a position-squeezed state? A momentum-squeezed state?

(c) Show that a squeezed state cannot be a coherent state (except for $r = 1$). How do their photon number distributions differ?

Section 4.8: Applications

Problem 4.29 [I]

Molecular spectroscopy. The fundamental vibrational frequency of ${}^{12}$C${}^{16}$O is $\nu_0 = 6.42 \times 10^{13}$ Hz.

(a) Calculate the spring constant $k$ of the CO bond.

(b) What wavelength of light is absorbed in the $v = 0 \to v = 1$ transition? In what region of the electromagnetic spectrum is this?

(c) The next overtone transition ($v = 0 \to v = 2$) is observed at almost exactly twice the frequency. Is this consistent with the harmonic oscillator model? What would anharmonicity do to this frequency?

(d) For ${}^{13}$C${}^{16}$O, predict the fundamental frequency. (Hint: only the reduced mass changes.)

Problem 4.30 [A]

From oscillators to fields. Consider a 1D chain of $N$ atoms, each of mass $m$, connected by springs of constant $k$, with periodic boundary conditions.

(a) Write the Hamiltonian and show that in terms of normal mode coordinates, it separates into $N$ independent harmonic oscillators.

(b) Find the normal mode frequencies $\omega_s$ as a function of mode index $s$.

(c) The zero-point energy of the chain is $E_0 = \sum_{s=1}^{N} \frac{1}{2}\hbar\omega_s$. For large $N$, convert this sum to an integral and discuss whether it converges or diverges. What does this tell you about the vacuum energy problem in quantum field theory?

(d) If each oscillator is in a coherent state $|\alpha_s\rangle$, show that the chain behaves as a classical wave in the limit of large $|\alpha_s|$.

Hints

Problem 4.8: Multiply two generating functions with different parameters $s$ and $t$, integrate over $\xi$ with the Gaussian weight, and compare powers of $s^m t^n$.
Problem 4.9b: The asymptotic form of Hermite polynomials for large $n$ near the turning points involves Airy functions.
Problem 4.16c: The proportionality constant involves $\sqrt{n!/(n-k)!}$.
Problem 4.19c: The ground state wavefunction is a Gaussian, which is exactly the minimum-uncertainty packet used in the derivation.
Problem 4.21: The classical turning point for the ground state is at $\xi_{\text{tp}} = 1$, so you need $\text{erfc}(1)/2$.
Problem 4.30a: Introduce $Q_s = \frac{1}{\sqrt{N}}\sum_j q_j e^{-2\pi i s j/N}$ as the normal mode coordinates.