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classes:2009:fall:phys4101.001:q_a_1005

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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.

I felt like the test covered the right range between regurgitation and doing things completely new - some material was already done but mostly it was a slight change from things we've already done. As far as the integrals in problem 1 go, it seems like since they have been in so many homework problems that you should be able to do them off the top of your head/on the fly, which seems like a bit of work to learn but really useful considering how many times we have done them and probably will end up doing them in the future.

I thought the test was okay even though I confused myself over the ladder operators and probably got a 0 on that question as a result. I'll agree with everyone else that the integrals shouldn't have been expected to be known since I'm pretty sure most of us just looked them up on an integral chart when we had similar ones for the homework. A hint for the next time they come up though, try switching the sin and cos terms into Euler's formula with <math>e^(i\theta)</math> terms as they are much more straightforward to integrate than things like sin squared.

On page 78 Griffiths says that κ is real. How can κ be real if E is positive? Can E be negative?

In the example here we are only considering bound states, which all have a negative E value by the definition of bound states (since the potential is defined as 0 and the bottom of the well is defined as <math>-V_0</math>). κ is only real in these situations, and for scattering states where the E is positive κ will have an imaginary term I believe.

classes/2009/fall/phys4101.001/q_a_1005.txt · Last modified: 2009/10/05 20:08 by yk