What is the De Broglie-Wave Problem in Special Relativity?

[fluidistic]

Homework Statement

Consider a particle whose rest mass is m_0. By analogy of E=h \nu for the electromagnetic field, de Broglie assumed that there existed some kind of intrinsic oscillatory motion with frequency \nu _0 associated to the particle at rest, where h \nu _0=m_0 c^2.
Assuming that the particle is moving with a velocity v with respect to an inertial frame of reference:
1)Show that for an observer in the fixed inertial reference frame the oscillatory motion of the particle is described by a progressive wave whose phase velocity is \frac{c^2}{v}.
2)Deduce the relation \lambda =\frac{h}{p}.
3)Show that the total energy of the particle satisfies E=h \nu in any intertial reference frame, where \nu=\gamma \n_0 and \gamma is Lorentz factor.

Homework Equations

Not sure.

The Attempt at a Solution

For 1) I should maybe find something of the form A \cos (bx+ct). But I really don't see how to even start. I'd like to solve 1) first and then proceed further.
I'd love a tip just to get me started... thank you very much.

 

[sgd37]

a de Broglie matter wave is of the form

e^{-i\vec{p} \cdot \vec{x}}

where

\vec{p} = (E, p_x , p_y , p_z ) and \vec{x} = (t , x ,y ,z)

so that you can see for one dimension it is a wave of the form e^{-i(\omega t - k x)}

the phase velocity is defined as v_p = \frac {\omega}{k} now use special relativity relations for the rest

 

[fluidistic]

Thanks for helping! I appreciate your time and help.

a de Broglie matter wave is of the form

e^{-i\vec{p} \cdot \vec{x}}

Nice to know, I never seen this before.

where

\vec{p} = (E, p_x , p_y , p_z ) and \vec{x} = (t , x ,y ,z)

So what are \vec p and \vec x? They seem like the momentum vector and the position vector but extended with energy and time? I never seen that either before. I'd like to know how do you call them.

so that you can see for one dimension it is a wave of the form e^{-i(\omega t - k x)}

I try to follow you on this but doing the dot product and considering only 1 dimension I get \vec p \cdot \vec x=(Et,xp_x). With the data of the problem I could simplify it to \vec p \cdot \vec x=(m_0c^2t,x\gamma m_0 v), unfortunately nothing looking like \omega t-kx.
Seems like I need further assistance.

the phase velocity is defined as v_p = \frac {\omega}{k} now use special relativity relations for the rest

Perfect.

 

[sgd37]

These are vectors commonly encountered in relativistic physics called the four momentum and the four dimensional space-time vector. Note that i have missed out constants of c so that the units of E and t should have the dimensions of momentum and space respectively.

The dot product of two vectors is defined as \vec{a} \cdot \vec{b} = (a_1 , a_2) \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = a_1 b_1 + a_2 b_2

in relativistic physics the dot product is defined differently. The difference being that the spatial parts have a minus in front of them

 

[fluidistic]

Thanks a lot! My bad I'm so rusty how could I forget that a dot product between 2 vectors gives a number and not a vector... ouch. :shy:

Ok so I reach \vec p \cdot \vec x =m_0 c^2t-x \gamma m_0 v. Setting \omega =m_0 c^2 and k=\gamma m_0 v, I get v_p=\frac{c^2}{\gamma v} instead of \frac{c^2}{v}.
Does this mean I should have considered the classical momentum p_x=m_0 v instead of the relativistic one p_x=\gamma m_0 v? I don't think so, thus I don't know what I did wrong.

 

[sgd37]

you're missing a factor of gamma in the energy term SR energy is given by E = \gamma m c^2 remember the particle is moving the energy relation you used is only true for a particle at rest

 

[fluidistic]

Ok good so that solves part 1).
I've been playing with equations for part 2) and I can't reach the answer.
I must deduce that \lambda =\frac{h}{p}. Therefore that h=\lambda p.

On one hand I have that E=\gamma h \nu _0=m_0c^2 \gamma \Rightarrow h=\frac{m_0 c^2}{\nu _0}.
On the other hand I have that \lambda p = \frac{2\pi}{k} \cdot \gamma m_0 v=2\pi which of course does not match the value of h. (Edit: Hmm now that I think, it might match the value of h but how to show this?)
I have made the use of the relation k=\frac{2\pi}{\lambda}. I can see no flaw in what I did, yet I do not get the result. Where did I go wrong?

 

[sgd37]

you just derived the relation you need

\frac{E}{p} = v_p

knowing that E = h \nu and v_p = \nu \lambda you can derive the wavelength momentum relation

 

[fluidistic]

Thank you, I solved part 2).
For part 3 I made a type in the latex formula, I forgot a n. I must show that E=h \gamma \nu_0; something I've been assuming till here. I feel like turning in circles. What should I assume, what to start with?

 

[sgd37]

you haven't assumed that at all the only thing you have assumed is that E = h \nu for arbitrary frequency now using the given assumption for a stationary particle E = h \nu_0 = mc^2 all you have to do is relate the energy of a stationary particle to that of a moving one

 

[fluidistic]

you haven't assumed that at all the only thing you have assumed is that E = h \nu for arbitrary frequency now using the given assumption for a stationary particle E = h \nu_0 = mc^2 all you have to do is relate the energy of a stationary particle to that of a moving one

Hmm I'm all confused. To me when you/I write \nu, I understand it as \gamma \nu _0. The same apply for m=\gamma m_0.
Seems like I shouldn't have thought this way?!
Just to be sure, when you write E = h \nu_0 = mc^2, do you mean E = m_0 \gamma c^2?

 

[sgd37]

yeah that is what i mean. They shouldn't really teach that special relativity stuff you don't see that gamma ever again after the first year