According to quantum field theory, an electron particle is an excitation of the electron field. Is it the waves of excitation in the electron field that are interfering in a double slit experiment?
Are these the same waves that are defined by the wave function solution of the Schrödinger equation? I thought the wave function was a probability wave that didn’t have any physical manifestation....
We need to tread cautiously here because in quantum field theory the quantum field is a purely mathematical object. It is an operator field i.e. a field whose value at every point in spacetime is a mathematical function.
However many (most?) physicists believe that the operator field is a description of some physical object in the same sense that the wavefunction of an electron is a description of the electron. So there is some physical electron field and the excitations of that field are the things we observe as electrons. But it is very important to remember that this is a rather hand wavy statement and that we have no idea what exactly the physical object is, or even if it really exists.
Having said this, the obvious physical interpretation is that for a non-interacting particle the states of this mysterious field are just products of the free particle wavefunctions. So if only a single particle is present then we observe the particle as the usual infinite plane wave. And this infinite plane wave diffracts as it passes through the slits and forms the interference pattern on the far side. Or I guess to be more precise the wavefunction $\psi$ diffracts, then when we measure the particle density $|\psi|^2$ we observe the diffraction pattern.
The best way, I think, at least, to understand quantum field theories is to start with a very simple one: phonons in a crystal lattice. In some sense, this is actually the only "really fully understood" kind of quantum field theory, because as soon as you talk about interacting quantum fields, the mathematics in continuous space no longer provides for a unique solution determined by the behavior over a limited range of scales.
In this system, we have an infinitely long chain or train of harmonic oscillators, notionally representing atoms in a one-dimensional crystal (like a metal like iron), at uniform spacing. Importantly, the oscillators are also coupled to each other, in that the Hooke's law restoring forces are provided by connections - which you can think of as like springs - to their adjacent oscillators.
And what you can find, then, when you work through the maths of this system, is that you can set up a quantum state for the oscillator chain that looks like a single oscillator has been "plucked", i.e. excited to its next higher vibrational level, while all the other oscillators remain at ground state (this state is not stable, of course, any more than its classical analog, i.e. a string of balls with springs between them where you've carefully stretched one ball out of equilibrium while holding the rest in place). That is, for each lattice position $i$ (where we imagine the atoms in the chain as numbered with integers), we can consider there to be a quantum state of the lattice $|i \uparrow \rangle$ where that that particular oscillator has been promoted to its first excited level, while the oscillators around it are still at ground level. And because these are separate and orthogonal quantum states for each index $i$, we can form nontrivial quantum superpositions. In particular, we can write a general state of the form
$$|\psi\rangle = \sum_{i=-\infty}^{\infty} \psi_i(i)\ |i\uparrow\rangle$$
with weighting factors $\psi_i(i)$. These weighting factors are suggestively labeled, because what this state suggests is that we have a "pluck" of a single oscillator, since that's what each $|i \uparrow \rangle$ going into the sum represents, yet with indeterminate position, given by the probability amplitudes encoded in $\psi_i$, that "fuzzify" (reduce information about) just which of the states is "meant", and thus fuzzify which lattice position is involved in the "plucking".
And that sounds an awful lot like a particle! It's the tiniest, most local excitation possible, but it is simultaneously fuzzed out with regards to where it is, in the same manner that an electron is a point-sized object, yet "where it is" is similarly ill-determined.
And we call that a phonon.
If we expand the lattice to two dimensions, so we can do something like the double-slit experiment, and then evolve the state, we will find the behavior is that $\psi_i$ spreads, just like the wave function of an electron does, and it is that, that interferes with itself.
So what is interfering with itself is the positional wave function of the phonon, exactly like how the electron's positional wave function interferes. And since the maths for electrons is almost entirely analogous if we ignore interactions at the field level, we can say that what is happening in that case is the same thing. The electron in QFT language is a point excitation of the Dirac field, that is then "fuzzed out" as to just which point is excited via a nontrivial positional wave function, which is also the electron's positional wave function. When it goes through the slits, that wave function interferes.
From the perspective of quantum mechanics, matter waves are interferring represented by schrodinger's wave function and represent probability amplitude whose superposition result in probability of electrons on screen.
Question is can particles could represented as wave, can two particles occupy same space as per superposition. What are dark fringes, why these fringes can't be occupied with any electron. For waves, these can be explained but for particles it maybe or not.
There is one way to have this interference with particles but that require stream of particles, not single emitting particle gun. As waves are result of movement of many particles and interaction of their effect. This is kind of statistics but precise, that is why it is not studied under statistics.
Suppose a particle gun is mounted and free to slide in lateral direction for recoil. There is two passage separated by a distance equivalent to product of distance between slits and the gun, and recoil travel of the gun. Now particles fire from the gun collide to the internal edges of slits and if again collide or not collide between the slits and the screen where they collected. Now they collide after slits depend upon how they approch to slits.
