de Broglie waves in Misner Thorne Wheeler exercise 2.1

[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

 

[Perturbation]

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

 

[pervect]

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.

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

?

 

[George Jones]

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

How, then, would you tackle the question?

 

[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.

 

[George Jones]

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.

Yesterday, I didn't have MTW at hand; today I do. Now I can see what your up against. 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, [itex]\hbar[/tex] times the phase is $p = \left( \hbar \omega , \hbar \vec{k} \right)$, 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<br /> <br /> $$p \cdot v = \left< \tilde{p} , v \right> \equiv \hbar \phi.$$<br /> <br /> This is my take on the presentation and question, which I find to be particularly unclear.[/itex]

 

[pervect]

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.

 

[syedamiriqbal]

I got the answer now

Thanks to all and especially to Goerge. I have tried my self the problem and foung that I was confusing v with velocity. Bad notation can kill you.

Amir