- #1

- 23

- 0

## Homework Statement

Consider an infinite square well defined by the potential energy function

U=0 for 0<x<a and U = ∞ otherwise

Consider a superposed state represented by the wave function ## \Psi(x,t)## given at time t=0 by

$$\Psi(x,0) = N \{(-\psi_1(x) + (1+ i)\psi_2(x)\}$$

1. Assume that the constant N is real and positive. Fix N such that Ψ(x, 0) is normalised to one particle.

2. Find Ψ(x, t) for all times t. Is your result correctly normalised? Is this state stationary?

3. In terms of {E

_{n}}, what is the average energy <E> of the superposed state? Does it depend on t?

4. Calculate <x> as a function of time t.

5. Given ##E_n=(\frac{\hbar^{2}}{2m})( \frac{n\pi}{a})^{2}## , find the average time for an electron to move back and forth once in a well 1 nm wide. Compare this with the classical result for an electron with energy equal to <E>.

## Homework Equations

## The Attempt at a Solution

I am really struggling with what the lecturer is asking here. I have an answer to 1. ##N=\frac{1}{\sqrt{3}}## then I am not sure where to go.

for 2. I think I get from the TDSE

$$ \Psi(x,0)=\frac{1}{\sqrt{3}} \{(-\psi_1(x)e^{-iE_1t/\hbar} + (1+ i)\psi_2(x) e^{-iE_2t/\hbar}\} $$

I am not sure if it is correctly normalised but It is not stationary as E

_{1}≠E

_{2}

Could someone please point me in the right direction. I have both Griffiths and Serway and have read through but cannot find anything that helps me understand what is being asked. If someone could either talk me through it or point to relevant sections in the text I would be very grateful.

Thanks