- #1
NeoDevin
- 334
- 2
Problem:
Find the momentum-space wave function \Phi_n(p,t) for the nth stationary state of the infinite square well.
Equations:
\Psi_n(x,t) = \psi_n(x) \phi_n(t)
\psi_n(x) = \sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)
\phi_n(t) = e^{-iE_n t/\hbar}
\Phi_n(p,t) = \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{-ipx/\hbar} \Psi_n(x,t) dx
Attempt:
\Phi_n(p,t) = \frac{\phi_n(t)}{\sqrt{a\pi\hbar}} \int^{\infty}_{-\infty} e^{-ipx/\hbar} \sin(\frac{n\pi}{a}x) dx
= \frac{\phi_n(t)}{\sqrt{a\pi\hbar}} \frac{1}{2i} \int^{\infty}_{-\infty}\Bigg(e^{i(\frac{n\pi}{a} - \frac{p}{\hbar})x} - e^{i(\frac{-p}{\hbar} - \frac{n\pi}{a})x}\Bigg) dx
= \frac{\phi_n(t)}{\sqrt{a\pi\hbar}} \frac{1}{2i} 2\pi \Bigg(\delta(\frac{n\pi}{a} - \frac{p}{\hbar}) - \delta(\frac{p}{\hbar} + \frac{n\pi}{a})\Bigg)
This doesn't seem right to me. Do I have this right, or am I missing something somewhere?
Find the momentum-space wave function \Phi_n(p,t) for the nth stationary state of the infinite square well.
Equations:
\Psi_n(x,t) = \psi_n(x) \phi_n(t)
\psi_n(x) = \sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)
\phi_n(t) = e^{-iE_n t/\hbar}
\Phi_n(p,t) = \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{-ipx/\hbar} \Psi_n(x,t) dx
Attempt:
\Phi_n(p,t) = \frac{\phi_n(t)}{\sqrt{a\pi\hbar}} \int^{\infty}_{-\infty} e^{-ipx/\hbar} \sin(\frac{n\pi}{a}x) dx
= \frac{\phi_n(t)}{\sqrt{a\pi\hbar}} \frac{1}{2i} \int^{\infty}_{-\infty}\Bigg(e^{i(\frac{n\pi}{a} - \frac{p}{\hbar})x} - e^{i(\frac{-p}{\hbar} - \frac{n\pi}{a})x}\Bigg) dx
= \frac{\phi_n(t)}{\sqrt{a\pi\hbar}} \frac{1}{2i} 2\pi \Bigg(\delta(\frac{n\pi}{a} - \frac{p}{\hbar}) - \delta(\frac{p}{\hbar} + \frac{n\pi}{a})\Bigg)
This doesn't seem right to me. Do I have this right, or am I missing something somewhere?
Last edited: