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OK, here is the problem:
An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is):
[tex]\phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}][/tex]
where [tex]E_1 = \hbar\omega_1 = \frac{\pi^2 \hbar^2}{2mL^2}[/tex], and [tex]E_2 = \hbar\omega_2 = \frac{4 \pi^2 \hbar^2}{2mL^2}[/tex]
First, I need to find the expectation value <x> for the wavefunction.
It seems to me that I need to multiply the wavefunction by its complex conjugate, put x in the middle, and integrate from -infinity to infinity. But attempting to do this yields some nasty integrals, like [tex]\int x sin^2 (\frac{\pi x}{L})[/tex] and [tex]\int sin(\frac{\pi x}{L})sin(\frac{2 \pi x}{L})e^{it(\omega_2-\omega_1)}[/tex] ; am I just doing this completely wrong or do I need to plow through the integrals?
An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is):
[tex]\phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}][/tex]
where [tex]E_1 = \hbar\omega_1 = \frac{\pi^2 \hbar^2}{2mL^2}[/tex], and [tex]E_2 = \hbar\omega_2 = \frac{4 \pi^2 \hbar^2}{2mL^2}[/tex]
First, I need to find the expectation value <x> for the wavefunction.
It seems to me that I need to multiply the wavefunction by its complex conjugate, put x in the middle, and integrate from -infinity to infinity. But attempting to do this yields some nasty integrals, like [tex]\int x sin^2 (\frac{\pi x}{L})[/tex] and [tex]\int sin(\frac{\pi x}{L})sin(\frac{2 \pi x}{L})e^{it(\omega_2-\omega_1)}[/tex] ; am I just doing this completely wrong or do I need to plow through the integrals?
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