#### Tom Mattson

Staff Emeritus

Science Advisor

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Here is a little puzzle that I'm sure I should know the answer to, but my brain is failing me.

Consider a particle with moving with speed [itex]u[/itex] along the x-axis in some frame S. So its (relativistic) momentum is [itex]p_x=\gamma_umu[/itex]. Its DeBroglie wavelength is [itex]\lambda=h/p[/itex].

Now consider the same particle in a frame S' which moves with speed [itex]v[/itex] relative to S in the x-direction. One might expect (naively perhaps) that the DeBroglie wavelength would be length-contracted according to:

[tex]\lambda '=\frac{\lambda}{\gamma_v}[/tex].

Now note that the momentum of the particle in S' is [itex]p_x'=\gamma_v(p_x-vE)=\gamma_v\gamma_um(u-v)[/itex]. If that transformed momentum is substituted into the DeBroglie relation, we get:

[tex]\lambda '=\frac{h}{\gamma_v\gamma_um(u-v)}[/tex].

Clearly there is a conflict. I'm inclined to say that the second formula is correct because it gives an infinite result if S' is boosted to the rest frame of the particle, which is what I would expect.

So the question is, which part of the above reasoning is wrong?

Consider a particle with moving with speed [itex]u[/itex] along the x-axis in some frame S. So its (relativistic) momentum is [itex]p_x=\gamma_umu[/itex]. Its DeBroglie wavelength is [itex]\lambda=h/p[/itex].

Now consider the same particle in a frame S' which moves with speed [itex]v[/itex] relative to S in the x-direction. One might expect (naively perhaps) that the DeBroglie wavelength would be length-contracted according to:

[tex]\lambda '=\frac{\lambda}{\gamma_v}[/tex].

Now note that the momentum of the particle in S' is [itex]p_x'=\gamma_v(p_x-vE)=\gamma_v\gamma_um(u-v)[/itex]. If that transformed momentum is substituted into the DeBroglie relation, we get:

[tex]\lambda '=\frac{h}{\gamma_v\gamma_um(u-v)}[/tex].

Clearly there is a conflict. I'm inclined to say that the second formula is correct because it gives an infinite result if S' is boosted to the rest frame of the particle, which is what I would expect.

So the question is, which part of the above reasoning is wrong?

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