De broglie wavelength and lorentz contraction

[serp777]

Am I correct in thinking that the quantum mechanical de broglie wavelength explains relativity's contraction of matter? because lambda = h/p, as the velocity of say a proton increases, the momentum also increases, and the wavelength should get smaller because lim p-->infinity of h/p = 0. At very high velocities near light, the distance between amplitude peaks should shrink very close to the Planck length, and appear as a wall.

 

[Jano L.]

Qualitatively, the de Broglie wavelength indeed shrinks when particle increases velocity. However, this has nothing to do with relativity, because actually the formula for length contraction is

$$l_{contracted} = \frac{l_0}{\gamma},$$

where $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$. When the rod is at rest, the rod still has length, $l_0$, which is a finite number.

The de Broglie formula gives

$$\lambda = \frac{h}{\gamma m v},$$

which is different function of velocity than the length in Lorentz contraction. Also, there is no "proper length" which the particle would have at rest. Rather, if the particle is at rest, there is no wave, or in other words, the wavelength is infinite.

The de Broglie formula is valid even in non-relativistic wave mechanics; so it seems this shrinking of the wavelength is not relativistic phenomenon.

 

[PhilDSP]

I think you would need to consider the effects of both $\lambda$ wavelength and frequency changes with velocity to arrive at the cumulative effect on both length and time metrics.