Matter Waves: What Does Frequency Represent?

[snoopies622]

I asked a question very closely related to this a couple months ago here

https://www.physicsforums.com/showthread.php?t=319693

so I'm not sure if this should be a new thread or a continuation of the old one. In any case, one of the premises used in constructing the Schrodinger equation is that the relation

$$E = h \nu$$

holds not only for electromagnetic radiation but for matter as well. My question is - when it comes to matter, what does $\nu$ refer to? What is oscillating at frequency E/h ?

 

[jtbell]

What is oscillating at frequency E/h ?

The quantum-mechanical "wave function", $\Psi(x,t)$.

 

[dx]

In quantum mechanics, the wave-functions which correspond to particles with definite momentum and energy are plane waves, i.e. functions of the form $$\psi(x,t) = A\exp[-i(kx - \omega t)]$$. As you can see, such plane waves are characterized by the two numbers k and ω called the wave number and angular frequency respectively. These are related to the energy and momentum of the particle by the formulas

$$E = \hbar \omega$$

$$P = \hbar k$$

So the wave-function of these particles can be written as

$$\psi(x,t) = A\exp[-\frac{i}{\hbar}(Px - Et)]$$

Px - Et is a relativistic invariant whose Lorentz covariant expression is P_μX^μ, where P_μ is the four-momentum (E,P).

 

[javierR]

What is oscillating at frequency E/h ?

Matter particles as well as photons have an associated frequency, so something is oscillating. In the case of photons, we're used to wavelike properties when they come in groups we know as light waves. But unlike these familiar waves, particles in quantum mechanics are described by probability waves, which previous posts called the wave-function. Rather than trying to think (erroneously) about the particle as a ball vibrating around at frequency E/h, you have to de-focus your mental image a bit and imagine something that simply does not have a single point location, but rather spread over space in the distribution of a wave and that also oscillates in time. Then the amplitude of this wave is directly related to the probability distribution in space of where the particle would be located if it began interacting classically (e.g. when it interacts with most macroscopic systems). The oscillating frequency of the probability wave is the E/h in your question. Until the classical-like behavior starts showing up, the wavelike properties of photons and matter particles are indeed physical in that they interact with each other like waves would interfere with each other when they meet. While the quantum nature of particles is often lost when they start forming "large" systems, classical light waves still carry the wavelike nature of their photon constituents.

 

[snoopies622]

This has been helpful. Introductory physics texts always mention the diffraction experiments in the late 1920's that confirmed the de Broglie hypothesis for electron wavelength, but I don't remember seeing one say anything about directly observing an electron frequency. I understand the meaning of the complex probability wave $\psi$ oscillating with time, but since this interpretation of the wave function wasn't published until 1926, I cannot help but wonder exactly what de Broglie himself had in mind. Does it matter anymore?