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I see in Griffiths (and in class) how we determined <math>v_g = \frac{d\omega}{dk}</math>. But how do we get <math>v_p = \frac{\omega}{k}</math>?
This is simply the relationship between frequency <math>f</math>, wavelength <math>\lambda</math>, and velocity <math>v</math> for any wave (<math>v=f\lambda</math>), written in slightly different terms. Instead of <math>f</math> and <math>\lambda</math> we have the wavenumber <math>k=\frac{2\pi}{\lambda}</math> and the angular frequency <math>\omega=2\pi f</math>.
The figure 2.10(b) in Griffiths is not correct because <math>\phi(k)</math> should be negative for some k, am I right?
Yuichi You are right. I never noticed this before.
What did everyone think of the test?
I think it was very good except for problem 1. I feel it is unfair to expect us to be able to do those integrals on the fly. Other than that, the professor did a very good job making a closed-book test that did not rely on memorization. Although I didn't do well, I walked out of the test feeling that it was my fault for not studying enough, not because the test was poorly-written. I think anyone who thoroughly went over the major proofs covered and the problems we were assigned should have been able to do well on all the problems except 1.
I agree with nikif002 totally, the integrals for 1 was out of the blues. Besides that, it was a fair test.
it might have helped to use the expression for x and p with the a+/a- operator, then you would not have to do integrals! and even though i forgot this on test myself, we should all remember it for the next test.
The ladder operators only work for the Simple Harmonic Potential, and we were dealing with the infinite potential well. But that is a good idea know the less, construct a raising an lowering for a infinite square well and see what happens.