What is the relationship between rest energy and frequency in quantum mechanics?

[LarryS]

In QM, the energy of a particle with a non-zero rest mass is Planck's constant times the frequency. How does the energy associated with the rest mass m₀c² fit into this picture? Planck wrote of an "instrinsic frequency" associated with the rest mass. Thanks in advance.

 

[daschaich]

In QM, the energy of a particle with a non-zero rest mass is Planck's constant times the frequency. How does the energy associated with the rest mass m₀c² fit into this picture?...

Well, setting
$$mc^2 = h\nu,$$
we can express the frequency in terms of the mass:
$$\nu = \frac{mc^2}{h},$$
which was first proposed by Louis de Broglie in 1924. Actually, de Broglie derived
$$\nu = \frac{\gamma mc^2}{h} = \frac{mc^2}{h\sqrt{1 - \frac{v^2}{c^2}}},$$
which reduces to the previous expression for $$v = 0$$. More information at Wikipedia.

 

[LarryS]

Well, setting
$$mc^2 = h\nu,$$
we can express the frequency in terms of the mass:
$$\nu = \frac{mc^2}{h},$$
which was first proposed by Louis de Broglie in 1924. Actually, de Broglie derived
$$\nu = \frac{\gamma mc^2}{h} = \frac{mc^2}{h\sqrt{1 - \frac{v^2}{c^2}}},$$
which reduces to the previous expression for $$v = 0$$. More information at Wikipedia.

I guess I'm still a little bit confused. So, a particle with a rest mass (m₀>0) which is at rest (v = 0) has a non-zero frequency and a non-zero wave length and therefore has a phase velocity while the particle is at rest...?

 

[daschaich]

I guess I'm still a little bit confused. So, a particle with a rest mass (m₀>0) which is at rest (v = 0) has a non-zero frequency and a non-zero wave length and therefore has a phase velocity while the particle is at rest...?

Sounds like it. In fact, the Wikipedia article I mentioned above links to this one, where we learn that the phase velocity of a massive particle always exceeds the speed of light c: "The superluminal phase velocity does not violate special relativity, for it doesn't carry any information."

The article goes on to contrast the phase velocity with the group velocity and the particle velocity. The particle velocity of a massive particle must be less than c, and is what vanishes for a particle at rest.