Introduction to Quantum Mechanics · Problem 6.30

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · quantum_propagator

Problem

Consider a time-independent Hamiltonian for a particle moving in one dimension that has stationary states

\psi_n(x)

with energies

E_n

. (a) Show that the solution to the time-dependent Schrödinger equation can be written

\Psi(x,t) = \hat{U}(t)\,\Psi(x,0) = \int K\!\left(x,x',t\right)\Psi\!\left(x',0\right)dx',

where

K\!\left(x,x',t\right)

, known as the propagator, is

K\!\left(x,x',t\right) = \sum_n \psi_n^*\!\left(x'\right)e^{-iE_n t/\hbar}\,\psi_n(x). \tag{6.79}

Here

\left|K\!\left(x,x',t\right)\right|^2

is the probability for a quantum mechanical particle to travel from position

x'

to position

x

in time

t

. (b) Find

K

for a particle of mass

m

in a simple harmonic oscillator potential of frequency

\omega

. You will need the identity

\frac{1}{\sqrt{1-z^2}}\exp\!\left[-\frac{\xi^2+\eta^2-2\xi\eta z}{1-z^2}\right] = e^{-\xi^2}e^{-\eta^2}\sum_{n=0}^{\infty}\frac{z^n}{2^n n!}H_n(\xi)\,H_n(\eta).

\Psi(x,t)

if the particle from part (a) is initially in the state

\Psi(x,0) = \left(\frac{2a}{\pi}\right)^{1/4}e^{-a(x-x_0)^2}.

Compare your answer with Problem 2.49. *Note*: Problem 2.49 is a special case with

a = m\omega/2\hbar

. (d) Find

K

for a free particle of mass

m

. In this case the stationary states are continuous, not discrete, and one must make the replacement

\sum_n \to \int_{-\infty}^{\infty} dp

in Equation 6.79. (e) Find

\Psi(x,t)

for a free particle that starts out in the state

\Psi(x,0) = \left(\frac{2a}{\pi}\right)^{1/4}e^{-ax^2}.

Compare your answer with Problem 2.21.