- #1

- 1,026

- 0

## Homework Statement

For each of the following wave functions check whether they are eigenfunctions of the momentum operator, ie whether they satisfy the eigenvalue equation:

[tex] \hat{p} \psi(x) = p\psi(x) with \hat{p} = i \hbar \frac{\partial}{\partial x} [/tex] and p is a real number.

For those that are eigenfunctions, calculate the eigenvalue p, the expectation value〈[tex] \hat{p} [/tex]〉, and the standard deviation Δ[tex]\hat{p}[/tex] ̂.

(a)

[tex] \psi(x) = \sqrt{\frac{2}{L}} cos(\frac{x\pi}{L}) for -L/2 \leq x \leqL/2[/tex]

0 for x > L/2, x < -L/2

(b)

[tex] \phi(x)= \frac{1}{\sqrt{L}}e^{ikx} for 0 \leq x \leq L with k is real [\tex]

0 for x > L, x < 0

(c)

[tex] \chi (x) = 2xe^{-x} for 0 \leq x [/tex]

0 for x < 0

## Homework Equations

N/A

## The Attempt at a Solution

Okay, I have gone some way through the first part.

[tex] \hat{p} = -i \hbar \frac{\partial}{\partial} [/tex]

[tex] -i\hbar \frac{\partial}{\partial x}(\sqrt{\frac{2}{L}}cos\frac{x\pi}{L}) [/tex]

[tex] -i\hbar \sqrt{\frac{2}{L}} \frac{\partial}{\partial x}(cos\frac{x\pi}{L}) [/tex]

With Limits between L/2 and -L/2

[tex] -i\hbar \sqrt{\frac{2}{L}} (-\frac{\pi}{L}sin\frac{x\pi}{L})^{L/2}_{-L/2} [/tex]

[tex] -i\hbar \sqrt{\frac{2}{L}} ([-\frac{\pi}{L}sin\frac{L\pi}{2L}]-[-\frac{\pi}{L}sin\frac{-L\pi}{2L}]) [/tex]

[tex] -i\hbar \sqrt{\frac{2}{L}} ([-\frac{\pi}{L}sin\frac{\pi}{2}]-[-\frac{\pi}{L}sin\frac{-\pi}{2}]) [/tex]

But I am unsure where I am supoosed to go from here...

To show it is a Eigenfunction, I need to get rid of the i at the very begining, but I am unsure how to do this?

Any suggesstions where to rpoceed?

Thanks in advance,

TFM