# Is any particle's energy quantized?

In the above picture, the author is trying to summarize the correlation between particle and wave packet. In doing so, he assumes that frequency is related to energy as: $E=h\nu$. Is this apparent assumption correct?

Because as far as I know, the quantization is only for oscillators on blackbody surface. And Einstein extended it for light waves.

But when it comes to matter wave, is it still true? That the particles energy can only be an integral multiple of the frequency associated with its matter wave times the Plank's Constant?

In fact, the slide doesn't even mention the integral multiple. So where am I missing out?

• That integral multiple is then interpreted as the number of particles with the energy corresponding to that frequency. This sounds trivial, but this is how one starts with setting up quantum field theory. In ordinary quantum mechanics one studies the low energy regime where the number of particles is fixed. In that case the energy of system follows from boundary condition on the wavefunction (matter waves). E.g. the energy levels of a hydrogen atom follow from the demand that the wavefunction must tend to zero at infinity. – Count Iblis Aug 28 '18 at 15:34

Any particle has an energy given by the relativistic equation for the energy:

$$E^2 = p^2 c^2 + m^2 c^4 \tag{1}$$

In this equation the variable $m$ is the rest mass and $p$ is the (relativistic) momentum. The momentum is related to the de Brogie wavelength by:

$$p = \frac{h}{\lambda} \tag{2}$$

For photons the rest mass is zero, so equation (1) for the energy simplifies to:

$$E = pc = \frac{hc}{\lambda} = h\nu$$

And that's why the energy of a photon is always equal to $h\nu$. With a massive particle the same equation applies, but with two big differences. Firstly the rest mass is no longer zero, and secondly the phase velocity is not $c$. So we can write the energy of the particle as:

$$E = \sqrt{p^2c^2 + m^2c^4} = \sqrt{\frac{h^2c^2}{\lambda^2} + m^2c^4}$$

which sadly is a rather less elegant expression.

However we can define a matter wave frequency:

$$\nu = \frac{E}{h}$$

and this automatically makes the energy equal to $h\nu$. Proceed with caution though as unlike light, with massive particles this frequency is not something that is directly measured. We can directly measure the de Broglie wavelength by diffracting the particles with some suitable grating, but the de Broglie frequency is rather more abstract.

• So that means, frequency of matter waves is defined like that? – Aditya Agarwal Aug 28 '18 at 18:52
• @AdityaAgarwal the frequency arises from the time dependent Schrodinger equation, so it isn't just a random definition. – John Rennie Aug 29 '18 at 5:33