Question

QUANTUM MECHANICS-upper level

In the harmonic oscillator problem, the normalized wave functions for the ground and first excited states are ψ0 and ψ1. Using these functions, at some point t, a wave function u = Aψ0 + Bψ1 is constructed, where A and B are real numbers.
(a) Show that the average value of x in the u state is generally non-zero.
(b) What condition A and B must satisfy if we want the function u to be normalized?
(c) For which values of the constants A and B do we get the maximum and for which minimum of <x>?

The result:
(b) A2 + B2 = 1
(c) Maximum: A = B = 2ˇ(−1 / 2)?

Answer 1

WE are given a wave function which is combination of ground and excited state

a) Say

In ladder operators

We also know

So the expectation value of x is

<u|x|u>=

<u|x|u>=

WE know that inner product of same states survive and of others vanish

Hence

<u|x|u>=

<u|x|u>=

Hence this expectation value is non zero

b) Let us normalise u

We know inner product of same state survives , other vanishes

For u to be normalised

Hence

This is the condition A and B must satisfy

c) To get maximum of <x>

<u|x|u>=

We need to maximize  

Also we have condition

            ----------(1)

Minimum will be when either A or B is zero

i.e A=1 B=0 or A=0 B=1

For maximum               -----------(2)

From 1 and 2

Put in eqn 1

At this value <x> will be maximum