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Problems & Solutions in Quantum Mechanics (II)

Problems 1032–1054 from Problems & Solutions in Quantum Mechanics (Lim, ed.)—harmonic oscillator, tunneling, scattering, and more.

Problem 1032Wisconsin

Harmonic Oscillator: General Solution, ⟨x⟩(t), and ⟨V⟩ = ½⟨E⟩

Problem Statement

A particle moves in a one-dimensional harmonic oscillator potential $V(x) = \frac{1}{2}m\omega^2 x^2$.

(a) Write down the most general solution $\psi(x,t)$ of the time-dependent Schrödinger equation.

(b) Show that $\langle x \rangle(t) = A\cos(\omega t) + B\sin(\omega t)$ for real constants $A, B$ determined by the expansion coefficients.

(c) Prove that the time-averaged expectation value of the potential energy equals half the total energy: $\overline{\langle V \rangle} = \frac{1}{2}\langle E \rangle$.

Step-by-Step Solution

Part (a): General Solution

The energy eigenstates of the harmonic oscillator satisfy $H\phi_n = E_n\phi_n$ with $E_n = (n + \tfrac{1}{2})\hbar\omega$. The most general solution is a superposition:

$$\psi(x,t) = \sum_{n=0}^{\infty} a_n \,\phi_n(x)\, e^{-iE_n t/\hbar}$$

where the coefficients $a_n = \langle \phi_n | \psi(x,0)\rangle$ are determined by the initial condition and satisfy $\sum_n |a_n|^2 = 1$.

Part (b): Expectation Value of Position

Using the ladder-operator representation $x = \sqrt{\hbar/(2m\omega)}\,(a + a^\dagger)$, the matrix elements of $x$ are nonzero only between adjacent states:

$$\langle \phi_m | x | \phi_n \rangle = \sqrt{\frac{\hbar}{2m\omega}}\bigl(\sqrt{n}\,\delta_{m,n-1} + \sqrt{n+1}\,\delta_{m,n+1}\bigr)$$

Therefore only $n \pm 1$ cross-terms contribute to $\langle x \rangle(t)$. The time-dependent phases give $e^{\pm i\omega t}$, so:

$$\langle x \rangle(t) = \sqrt{\frac{\hbar}{2m\omega}} \sum_{n=0}^{\infty} \sqrt{n+1}\bigl(a_n^* a_{n+1}\,e^{-i\omega t} + a_{n+1}^* a_n\,e^{i\omega t}\bigr)$$

Writing $a_n^* a_{n+1} = |a_n^* a_{n+1}|e^{i\phi_n}$ and combining:

$$\langle x \rangle(t) = A\cos(\omega t) + B\sin(\omega t)$$

where $A$ and $B$ are real constants depending on the coefficients $a_n$. The expectation value oscillates at the classical frequency $\omega$—an instance of Ehrenfest's theorem.

Part (c): Virial Theorem — Time-Averaged Potential Energy

For a stationary state $\phi_n$, the virial theorem for the harmonic oscillator gives $\langle T \rangle_n = \langle V \rangle_n = \frac{1}{2}E_n$. For a general state:

$$\langle V \rangle(t) = \sum_{n,m} a_m^* a_n \langle \phi_m | V | \phi_n \rangle \, e^{i(E_m - E_n)t/\hbar}$$

Time-averaging kills all cross-terms ($m \neq n$) since $\overline{e^{i(E_m - E_n)t/\hbar}} = 0$ for $E_m \neq E_n$. Only diagonal terms survive:

$$\overline{\langle V \rangle} = \sum_n |a_n|^2 \langle \phi_n | V | \phi_n \rangle = \sum_n |a_n|^2 \cdot \frac{1}{2}E_n = \frac{1}{2}\langle E \rangle$$

This is the quantum virial theorem for the harmonic oscillator: on average, kinetic and potential energies each carry half the total energy.

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