Physics: The Ultimate Adventure by Ross Barrett Pier Paolo Delsanto & Angelo Tartaglia

Author:Ross Barrett, Pier Paolo Delsanto & Angelo Tartaglia
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

8.7 Superluminal Phase Waves

Although QM has been developed in accordance with special relativity, some difficulties arise when one tries to reconcile the two theories. As we have seen in Chap. 7, a fundamental tenet of special relativity is that the speed of light c is the same for all inertial observers and that no physical object can travel faster than c. We have also seen that in Quantum Mechanics each particle is associated with a wave function and may be described in terms of simple harmonic waves: i.e. sines and cosines. A harmonic wave propagates with a velocity, called phase velocity , at which the crests (or the valleys) of the wave are traveling. The phase velocity is given by the product of the frequency times the wavelength. Since electromagnetic waves can be described as streams of photons, their phase velocity is given by the ratio between the energy of the photon (which is proportional to the frequency of the wave) and its momentum (which is inversely proportional to the wavelength). As a result, the speed of the photon turns out to be exactly c.

Let us apply the same argument to a free electron moving at a given speed V (lower of course than c) and with an associated quantum wave. The phase velocity is the ratio between the total relativistic energy of the particle mc 2 and its momentum mV. The result is larger than c for a subluminal particle (V less than c), which is consequently associated with a superluminal wave (i.e. phase velocity greater than c)! This result stresses the non-physical nature of the wave function, even if it is used to evaluate physical quantities.

If we consider a real electron, which occupies some finite portion of space, we describe it quantum-mechanically as a wave packet , i.e. as a superposition of ideal harmonic plane waves each having a different (superluminal) phase velocity. Looking for the group velocity of the wave packet, we correctly get the subluminal V, the physical velocity of the particle.

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