Stationary states infinite cubic well

[happyparticle]

Homework Statement

For which values of c the state $e^{c \cdot L_z} |2,2,2>$ is stationary for the infinite cubic well Hamiltonian.

Relevant Equations

$\psi (r,t) = \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}$
$E = \frac{6}{m} (\frac{\pi \hbar}{l})^2$

For a state to be stationary it must be time independent.

Naively, I tried to find the values of c where I don't have any time dependency.

$e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}$

$e^{c \cdot L_z -iEt/\hbar}$

Thus
$c = \frac{L_z - iEt}{\hbar}$

$c = \frac{L_z - i\frac{6}{m} (\frac{\pi \hbar}{l})^2t}{\hbar}$

I'm wondering if this is correct.

Thanks

 

[Philip Koeck]

Homework Statement:: For which values of c the state $e^{c \cdot L_z} |2,2,2>$ is stationary for the infinite cubic well Hamiltonian.
Relevant Equations:: $\psi (r,t) = \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}$
$E = \frac{6}{m} (\frac{\pi \hbar}{l})^2$

For a state to be stationary it must be time independent.

Naively, I tried to find the values of c where I don't have any time dependency.

$e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}$

$e^{c \cdot L_z -iEt/\hbar}$

Thus
$c = \frac{L_z - iEt}{\hbar}$

$c = \frac{L_z - i\frac{6}{m} (\frac{\pi \hbar}{l})^2t}{\hbar}$

I'm wondering if this is correct.

Thanks

A stationary state itself is not time independent. Its absolute square is time independent.
That won't get you very far though. You have to look at something else, I would say.

 

[happyparticle]

The eigenfunction is $\sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}$
The eigenvalue is $\frac{12 \hbar^2}{8mL^2}$
Maybe $e^{c L_z} = e^{\frac{12 \hbar^2}{8mL^2}}$

 

[vela]

I assume $L_z$ is the operator corresponding to the z-component of orbital angular momentum, so your attempts don't make mathematical sense. You should go back and review the basics of operators and wave functions.

If it's not supposed to represent that operator, what is it supposed to be?