A particle of mass in a harmonic oscillator potential is in its ground state

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch The wave function is not the state

Problem

A particle of mass

m

in a harmonic oscillator potential

V(x)=\frac12m\omega^2x^2

is in its ground state. In the position representation the normalised ground-state wave function is

\phi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\!\left(-\frac{m\omega x^2}{2\hbar}\right),

In the energy representation the state is simply

c_n(0)=\delta_{n0}

, i.e.

(1,0,0,\dots)

, with

E_n=(n+\tfrac12)\hbar\omega

. (a) Write

|\phi_0(x)|^2

as a Gaussian

\propto e^{-x^2/2\sigma_x^2}

and read off

\langle \hat x^2\rangle=\sigma_x^2

without doing an integral. (b) Fourier transform

\phi_0

\tilde\phi_0(p)=(2\pi\hbar)^{-1/2}\int_{-\infty}^{\infty}\phi_0(x)e^{-ipx/\hbar}\,\mathrm{d}x

, by completing the square in the exponent. Then do for

|\tilde\phi_0(p)|^2

what part (a) did for

|\phi_0(x)|^2

to get

\langle \hat p^2\rangle=\sigma_p^2

, and combine it with

\langle \hat x^2\rangle

from part (a) to compute

\langle\hat H\rangle = \langle\hat p^2\rangle/2m + \frac12m\omega^2\langle\hat x^2\rangle

. Compare with

E_0

read directly off the energy-basis list. (c) Substitute

\phi_0(x)

into the time-independent Schrodinger equation

-\frac{\hbar^2}{2m}\phi_0''(x)+\frac12m\omega^2x^2\phi_0(x) = E\phi_0(x)

and show by direct differentiation that it is satisfied with a specific value of

E

, confirming that this position-basis equation is the component form of the basis-free statement

\hat H|\psi\rangle=E_0|\psi\rangle

. (d) The full time-dependent state is

\Psi(x,t)=\phi_0(x)e^{-iE_0t/\hbar}

. State, without further calculation, the value of

P(E_0)

predicted from the energy representation and the value of

\int_{-\infty}^{\infty}|\Psi(x,t)|^2dx

predicted from the position representation, for every

t

, and say in one sentence why the two must agree.