Probability Waves vs. Amplitude Waves It is often asserted, and it is common knowledge, that the waves associated with a particle are probability waves.
This seems reasonable. But what about $E=hf$? This does not seem to be about probability.
Is it the case that sometimes we are dealing with one type of wave, and sometimes with another?
 A: Yes, there are two different types of waves involved here. In classical electrodynamics an electromagnetic wave describes how the magnitude and direction of the electric and magnetic fields vary with time and with position in physical space. In quantum mechanics a wave function describes how the complex-valued probability amplitude of a quantum system varies with time and with position in the system's configuration space, which is an abstract space of possible configurations of the system.
When we say that $E=hf$ for a photon, the frequency $f$ is both the frequency of the quantum wave function for that photon and the frequency of the electromagnetic wave which would classically represent a population of photons with the same energy $E$; in other words, the classical electrodynamic frequency and the quantum wave function frequency of a photon have the same value.
A: In this double slit one photon at a time experiment one sees the interference pattern coming from classical electromagnetic waves in the last frame on the right,



*

*Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.


The first frame shows the distribution of single photons that make up the classical interference. At the quantum level, the last frame is the probability distribution for  the experiment "photon scattering off double slit given width and dimensions" the $Ψ^*Ψ$ of those boundary conditions .
$E=hν$ is the energy of the photons used in the experiment, where $ν$ is the frequency of the classical wave
In one experiment two types of waves give the interpretation of the data. The reason the classical wave and the quantum probability wave have the same frequency you will find in the reference here, in a nutshell, because  quantized Maxwell equations are used to get the wave function of a photon, so the quantized equation  solutions are related to the classical equations.
A: The equation $E=h f$ is rather an experimental fact (from the blackbody radiation or the photoelectric effect). It says that the particles with energy $E$ have an associated wave with frequency $f$ and vice versa. But it is not really clear at this point that this wave is a probability wave. This probabilistic behaviour of particle can be well shown in the double slit experiment with electrons. Other experiments showed that this probabilistic wave should also obey the relation $p = \frac{h}{\lambda}$. From these two experimental fact we can say that a wave function with energy $E$ and momentum $p$ can be written as:
$\begin{align}\psi(x,t) = A \cdot e^{-i \cdot 2 \pi f t} \cdot e^{\frac{i \cdot 2 \pi}{\lambda}x}= A \cdot e^{-i \frac{ E }{\hbar}t} \cdot e^{i \frac{ p}{\hbar}x} \end{align}$
From this you can "derive" the Schrödinger equation etc. Scientists then realised that we could think of the square of this wave function as the probability for a particle to be measured at $x$ to the time $t$.
So this long story was to show you that $E = hf$ is an experimental fact that tells you how the probabilistic wave function evolves over time. It is not a consequence of the probabilistic behaviour.
