I am studying QM-I these days. Now, I just think of the wave function as just a mathematical function that defines the state of the particle at an instant and from it you can extract various observables. I have been told that it is just a misnomer calling it an actual wave, and there is no wave there, its just that the wave function satisfies the general wave differential equation in 3d space. Now is this true? Is there really no wave? What about QFT, when we say electron is an excitation in a Quantum Field, do we mean to say, it is sort of a pulse in that field medium? Like a photon is a pulse in an EM wave and essentially electron is just a pulse, which we see as a particle? Or is there no wave there ? Is this related to $E=\gamma mc^2$ by relating energy in a wave to a particle.
[Physics] Quantum Wave Mechanics
quantum mechanicsquantum-interpretationswave-particle-dualitywavefunction
Best Answer
Except in very elementary examples (single particles), the QM wave function has nothing to do with a wave (apart from the historical origin).
For a system consisting of $N>1$ particles, the wave function is a function in configuration space (with 3N variables), not one in 3-space (whose coordinates are positions $x$ with 3 components). This can be read in any textbook on quantum mechanics. Whatever oscillates in configuration space has (therefore) little to do with oscillations of waves in space and time.
In quantum field theory, one has true waves, which are oscillations of expectation values of field observables or their products. But these have nothing to do with wave functions either. Indeed, the analogue of a QM wave function in QFT is a wave functional, which are functions $\psi(\Phi)$ depending not on space position $x$ and time $t$ (as the wave function of a single particle) but on all fields $\Phi$ (which themselves depend on $x$ and $t$). These wave functionals are not easy to work with, so you don't find them even mentioned in introductions to QFT. A recent (but not elementary) reference is Phys. Rev. D 77.085007 (2008). A much older, but more readable reference is Phys. Rev. D 37 (1988), 3557-3581.
When we say that an electron is an elementary excitation in a quantum field, it has a similar meaning as when we say that a sine wave is an elementary excitation of a string. The spectrum of an ideal string consists of a ground frequency and its overtones - integral multiples of the frequencies. The spectrum of a real string (or plate, etc.) is more complicated but the ground frequency is still a well-defined mode.
The spectrum of the quantum field in its rest frame consists of a discrete spectrum and a continuous spectrum. The reason for a continuous spectrum is that multiple excitations may move in different directions (though the rest frame stays fixed), and their kinetic energy adds a continuous amount to the frequency $\nu$, according to the formulas $E=h\nu$ and $E=E_0+mv^2/2$ (for a nonrelativistic motion with rest energy E_0 and speed v). The discrete part of the spectrum is made up of modes which are called particles - by tradition, although their properties don't resemble much those of little balls.
Single modes of quantum fields fields are described by classical wave equations with space-time arguments, and hence have a natural (and classically describable) interpretation in terms of waves. These classical wave equations are quantized to get the quantum fields. But a mode of a quantum field is something very different from a wave functional: A wave functional is a superposition of tensor products of an arbirary number of modes; the modes are just their building blocks.