Quantum Mechanics and Eigenfunctions Checks

[TFM]

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:

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

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

(a)

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

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)

$$\chi (x) = 2xe^{-x} for 0 \leq x$$
0 for x < 0

Homework Equations

N/A

The Attempt at a Solution

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

$$\hat{p} = -i \hbar \frac{\partial}{\partial}$$

$$-i\hbar \frac{\partial}{\partial x}(\sqrt{\frac{2}{L}}cos\frac{x\pi}{L})$$

$$-i\hbar \sqrt{\frac{2}{L}} \frac{\partial}{\partial x}(cos\frac{x\pi}{L})$$

With Limits between L/2 and -L/2

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

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

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

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

 

[nrqed]

You do not need the boundary conditions to check if a function is an eigenstate so no need to replace x by L/2 or -L/2. These limits only enter when you calculate the expectation values.


You must ask yourself if the function you get after applying $i \hbar \partial_x$ to $\psi$ is proportional to the wavefunction itself. If yes, then the wavefunction is an eigenstate and the constant of proportionality is the eigenvalue. If not, the function is not an eigenstate.

 

[TFM]

So do I need to differnitiate it then?

 

[nrqed]

So do I need to differnitiate it then?

Yes. Sorry, there was a typo in my post and the partial derivative did not show up. Yes, you must apply $i \hbar \partial_x$ to $\psi$ and check if the result is proportional to the wavefunction itself. Is it possible to write $\sin(\pi x/l)$ as a constant times $\cos (\pi x/L)$ in such a way that the result will be valid for any value of x ? If the answer is no, then this wavefunction (the cos) is not an eigenstate of the momentum operator

 

[TFM]

Okay so the original wavefunction is:

$$\psi(x) = \sqrt{\frac{2}{L}} cos(\frac{x\pi}{L})$$ --- (1)

The $$\hat{p}$$ function is:

$$-i\hbar \sqrt{\frac{2}{L}} (-\frac{\pi}{L}sin\frac{x\pi}{L})$$ Which can be simplified to:

$$i\hbar \sqrt{\frac{2}{L}} * \frac{\pi}{L}sin\frac{x\pi}{L}$$

So now I have to see if this result is proportional to (1).

So I have to find how to write $$sin(\frac{\pi x}{L})$$ as $$C cos(\frac{\pi x}{L})$$

Can this actually be done, though, since the only relation I know that will link sin and cos is $$sin^2(\theta) + cos^2(\theta) = 1$$?

Also, do I need to get rid of that i at the beginning, since p needs to be a real number?