Lecture 11 Problem solving

Problem 1

A particle of mass m in the harmonic oscillator potential starts out in the state

for some constant A.

(a) What is the expectation value of the energy?

(b) At some later time T the wave function is

for some constant B. What is the smallest possible value of T?

Solution

First, let's introduce standard notations for harmonic oscillator:

This function can be expressed as a linear combination of the first three states ofharmonic oscillator.

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Now, we need to find coefficients c by equating same powers of

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Normalization gives:

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Now it is really easy to find the expectation value of energy:

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Problem 2

(a) Show that the wave function of a particle in the infinite square well returns to its

original form after a quantum revival time

for any state (not just a stationary state).

(b) What is the classical revival time, for a particle of energy E bouncing back and forth between the walls?

(c) For what energy are the two revival times equal?

Solution

The most general solution for the infinite square well potential is:

Therefore,

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(b) The classical revival time is the time that particle travels from one side of the well to the other and back.

(c) Quantum and classical revival times are equal if

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Problem 3

Let Pab(t) be the probability of finding a particle in the range ( a<x<b), at time t.

The is called probability current since it tells you the rate with which probability is"flowing" past the point x. its units?

(b) Find the probability current for the wave function

Solution

What are

In one of the lectures, we found that

Comparing it with definition of

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Probability is dimensionless, so J has dimensions 1/time, and

units (seconds)-1.

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Note on the calculation of integral in Homework #4.

Change of variables

Therefore,

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