In quantum mechanics, what is the difference between group velocity and phase velocity of matter wave? How can it also be that phase velocity of matter wave always exceeds the speed of light?


I will not go into math (because many already did ) but will try to explain physics.

Wave function describes the wave of probability (this of course rough analogy) it means when it travels in space, information of probability (probability of finding the particle) travels with it, thus this is phase velocity (which can have higher speed than speed of light) and it is not a real "physical matter" wave, but just a flow of information on probability, but it still a special kind of waves, in sense it can't have additional cargo of informations that may enable us to pass any additional useful information, thus it will not violet relativity in any way, while group velocity describes wave packages speed (also very rough and not very true analogy, but very helpful to build intuition) and this package can be imagined as your particle, thus it has the same speed of your particle.


Actually matter wave describes the probability of finding a particle at any time at any point of space. It is not exactly a sine wave, it is an wave packet. So it contains lots of component of single frequency wave. These the velocity of the single component is called phase velocity. And the overall velocity of composite wave is called the group velocity.

The phase velocity actually don't carry any meaningful physical information for matter wave. All the meaningful information (like, momentum, velocity etc) contains in the group wave. So the velocity of the particle is determined by the group velocity, not phase velocity.

Now since phase velocity does not carry any physical signal, so it can have speed greater than light. It does not violet the theory of relativity. Only the group velocity should be less than speed of light.

Hope this will help you understand


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