# Ex 2.1 in Misner Gravitaition

1. Jun 3, 2006

### syedamiriqbal

HI

I am trying to work through all the exercises in MTW. A very easy problem Exercise 2.1 about de broglie waves is not solved by me although it seems to be very simple. Could anyone help me out?

Amir

2. Jun 3, 2006

### Perturbation

It would help if you gave the exercise. Misner et al is quite an expensive book, I certainly can't afford it.

3. Jun 3, 2006

### pervect

Staff Emeritus
I'm not sure what your problem with the problem is.

What is the a) magnitude and b) direction of the momentum of the particle with the specified wavefunction.

$$\psi = exp[i(k x - \omega t)]$$

?

4. Jun 3, 2006

### George Jones

Staff Emeritus
Pretend that the question is in a modern physics or quantum mechanics text.

How, then, would you tackle the question?

5. Jun 4, 2006

### syedamiriqbal

The problem is to prove that p.v=<p.v> for the phase of de broglie wave as quoted by pervect above, where p on rhs is a one form and others are 4-vectors.

Last edited: Jun 4, 2006
6. Jun 4, 2006

### George Jones

Staff Emeritus
Yesterday, I didn't have MTW at hand; today I do. Now I can see what your up against. :grumpy: I like MTW very much, but I dislike the presentation in this part of the book - body piercings (I'm too old for that sort of stuff), bongs of bell, etc.

Given a 4-vector $p$, $p \cdot v = \left< \tilde{p} , v \right>$ for all 4-vectors $v$ is the *definition* of $\tilde{p}$, and one doesn't go around proving definitions, notwithstanding the stuff written on page 58.

I think you're just supposed to note that, in a particular frame, $\hbar[/tex] times the phase is [itex]p = \left( \hbar \omega , \hbar \vec{k} \right)$, and the 4-position is $x = \left( t , \vec{x} \right)$. The arbitrary 4-position plays the role of the arbitrary 4-vector $v$, so that (2.14) is

$$p \cdot v = \left< \tilde{p} , v \right> \equiv \hbar \phi.$$

This is my take on the presentation and question, which I find to be particularly unclear.

Last edited: Jun 4, 2006
7. Jun 4, 2006

### pervect

Staff Emeritus
I don't find the "bongs of the bell" approach all that bad - it seems very intuitive to me. But I gather that it drives people who are more mathematically rigorous crazy. The solution seems to me to just not be that rigorous.

We can divide the above problem into two parts. The first part is the quantum-mechanical part. The answer to that part of the problem is that one computes the component of momentum in the 'x' direction, and multiplies that by the component of the velocity in the 'x' direction, and that is the answer to that part of the problem.

The second part of the problem is to find out how many surfaces of constant phase the velocity vector passes through. This relates to a particular geometric inteprretation of the one-form as a set of "stacked plates". You then multiply this number by a constant, hbar, and show that this is the same as the result of the quantum-mechanical solution. That's all that's being asked.

8. Jun 6, 2006