- #1
painfive
- 23
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This is a problem from my introductory quantum mechanics class. It's Griffifth's problem 2.6, if anyone has that book. The problem says to investigate the effect of adding two steady state solutions with a relative phase. Namely:
[tex]\Psi(x,0) = A [ \psi_1(x) + e^{i \phi} \psi_2 (x) ][/tex]
where:
[tex]\psi_n(x) = \sqrt{\frac{2}{a}} \sin \left(\frac{n \pi}{a} x\right)[/tex]
He says to compare this to the case with no relative phase, and to study the special cases of phi=pi/2 and pi. But I get that the only difference is that it effectively shifts the time origin. For example, I get:
[tex]|\Psi(x,t)|^2 = \frac{1}{a}\left[\sin^2\left(\frac{\pi x}{a}\right) + \sin^2\left(\frac{2\pi x}{a}\right) + 2\sin\left(\frac{\pi x}{a}\right) \sin\left(\frac{2\pi x}{a}\right) \cos\left(\frac{3 \pi^2 \hbar t}{2 m a^2} - \phi\right) \right][/tex]
This isn't anything that needs to be investigated, and the special cases look meaningless, which makes me think I've done something wrong. Have I?
[tex]\Psi(x,0) = A [ \psi_1(x) + e^{i \phi} \psi_2 (x) ][/tex]
where:
[tex]\psi_n(x) = \sqrt{\frac{2}{a}} \sin \left(\frac{n \pi}{a} x\right)[/tex]
He says to compare this to the case with no relative phase, and to study the special cases of phi=pi/2 and pi. But I get that the only difference is that it effectively shifts the time origin. For example, I get:
[tex]|\Psi(x,t)|^2 = \frac{1}{a}\left[\sin^2\left(\frac{\pi x}{a}\right) + \sin^2\left(\frac{2\pi x}{a}\right) + 2\sin\left(\frac{\pi x}{a}\right) \sin\left(\frac{2\pi x}{a}\right) \cos\left(\frac{3 \pi^2 \hbar t}{2 m a^2} - \phi\right) \right][/tex]
This isn't anything that needs to be investigated, and the special cases look meaningless, which makes me think I've done something wrong. Have I?