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- 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

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