We know that when things move close to the speed of light, their lengths contract. Can this be observed when we accelerate electrons close to the speed of light? Do they measure shorter when they're at that speed?
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4$\begingroup$ what "length " do you think electrons have? $\endgroup$– ACuriousMind ♦Commented Feb 28 at 0:35
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3$\begingroup$ Electrons are point particles, so they do not have any length! $\endgroup$– vengaqCommented Feb 28 at 0:58
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1$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Feb 28 at 1:57
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2 Answers
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As far as we know experimentally, electrons are point particles, so they don't have a "length" per se.
However, quantum mechanically, there is uncertainty in the position of a point particle. For a single particle, we can describe this uncertainty as a wavefunction. The wavefunction is like a wave mathematically, except instead of being "ripples in a medium", the wavefunction represents the probability to discover a particle at a given location. A simple kind of wavefunction is a sine wave, and in this case the wavelength of the wave $\lambda$ is related to the particle's momentum $p$ and Planck's constant via the famous equation $p = h / \lambda$. The wavelength can be observed directly in interference experiments like the double slit experiment, for example.
From the formula $p = h/\lambda$, we see that as the momentum increases, the wavelength decreases. This is exactly what you would expect based on special relativity. As you boost into a frame where the particle has a larger momentum, the wavelength should shrink due to length contraction. Even though the formula we wrote down is non-relativistic in that it does not involve $c$, it can be made relativistic in the way we relate momentum and energy (or wavelength and frequency). The fact that the formula is compatible with length contraction is not an accident and part of how it is possible to reconcile special relativity and quantum mechanics in the framework of relativistic quantum field theory.
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Note also that relativistic velocities affect the shape of the electric field in space created by a moving charge, since that field propagates at c.
answered Feb 28 at 3:20