Probability and double slit

Question

if a beam of identical particles at random distances from each other (or exactly 1/2 lambda between each other) travelling with the same v towards a double sllit do not interfere with each others wave function, so that the wave function of each particle upon reaching the double slit is always lambda / p, thus producing a predictable interference pattern,

how come the wave function of the region BETWEEN the slits DOES interact with the wavefunction of each particle so as to 'block' (or greatly reduce) it so that each particle's wave function can interfere with ITSELF via the two slits?

conversely, since the wavefunction of one particle (the inter-slit 'substance') can and seemingly does affect the wavefunction of another 'particle' (that of the approaching particle), why don't the particles in a beam interfere with each other so as to randomoly destroy any assignable wavelength to them?

if the inter-slit region had no affect on the wavefunctions of the approaching particles, the diffraction grating / double slit apparatus would be entirely transparent to the beam and there would be no interference at all.

The probability function is an entirely mathematical construct, and yet how it evolves over space must be dependent on whether there is any 'matter' in the region through which it passes. Is there something like a "damping factor" of freespace, for the probability function, like electrical permitivity of freespace?

Answer 1

I think you have a few misconceptions here. You start by talking about the particles in the beam "not interfering with each other" so the "wave function of each particle is lambda/p".

There are at least two problems with this statement. I'll take the last part first. It looks like you are confusing "wave function" with "wave length". The wave function doesn't have a value. It is a function. In a case like this we would write it as a function of position and particle momentum. We would find this function by solving the Schrodinger equation, but let's not get into how that is done.

Next is not useful to think of the particles "interfering" with each other. A wave function of a single particle "interferes with itself" but wave functions of different particles don't interfere with each other. The wavefunction of each particle needs to be solved for from the Schrodinger equation. Without going into gory detail about how that is done, it depends on what the interactions are with the rest of the system (in this case, interactions with the slits and with the other particles).

Essentially your question is "why do the slits affect the wavefunctions of the particles but the other particles don't". There are two answers:

1. Typically the distances between particles in the beam are so large that the interactions between the particles are very weak and can be neglected.

2. Even if the interactions are not very weak (a very high intensity beam) then each particle in the beam has many particles around it (in all directions). Typically, these particles are all charged. For sake of argument let's say that they are electrons. So they all repel each other. But that means that the repulsion due to all of the nearby electrons will tend to cancel out and once again, at least for most of the electrons, we can ignore the interactions.

An accelerator physicist could answer this better, but I'm guessing that reason 1. is more important most of the time.

Answer 2

The probability function is an entirely mathematical construct, and yet how it evolves over space must be dependent on whether there is any 'matter' in the region through which it passes. Is there something like a "damping factor" of freespace, for the probability function, like electrical permitivity of freespace?

Some terms used in physics refer to mathematical tricks for doing calculations, like the centre of mass. You can do calculations and understand the motion of particles without invoking the centre of mass.

You can't make predictions in quantum mechanics without invoking the state. So the state is not a fiction for doing calculations. Rather, the wave function represents how the world really works according to the theory. The only way to dismiss it as a mathematical trick without shooting yourself in the foot is to invent an alternative to quantum mechanics that has no wave functions and makes independently testable predictions that disagree with those of quantum mechanics.

The particle you detect when you do a measurement is a particular kind of state that is highly peaked in position and momentum space on the scale of a person. On a small enough scale, this state is not approximately localised and doesn't bear much similarity to an object in a particular place moving on a particular trajectory. Rather the state is a sort of blob in position and momentum space and you can't understand the amplitude of the blob at a particular place over time without referring to the properties of the blob over a larger region as a result of interference. See 'The Fabric of Reality' by David Deutsch chapter 2 and the 'The Beginning of Infinity' by Deutsch, chapters 11 and 12 for more explanation.

If you're dealing with the state of an electromagnetic field, one of the properties of that state is the photon number. If you have a photon number of two in some particular region those two photons don't interact with one another directly. But the field's interaction with the barrier can affect the photon number because the barrier absorbs photons. The lack of interaction between two identical photons and the interaction between photons and the barrier are both reflected in the equations of motion for the field and the barrier.

The number of electrons is also a property of the electromagnetic field. If you had two electrons going through the slits at the same time, they would interact to some extent: they would repel one another. They would also be absorbed the barrier. Again, those interactions would be reflected in the equations of motion of the electromagnetic field and the barrier.

If you want to understand quantum field theory, a good place to start is 'Quantum field theory for the gifted amateur' by Lancaster and Blundell.

Answer 3

how come the wave function of the region BETWEEN the slits DOES interact with the wavefunction of each particle so as to 'block' (or greatly reduce) it so that each particle's wave function can interfere with ITSELF via the two slits?

The difference lies in the concept of superposition of wave functions and a single wave function for the given boundary conditions.

In Quantum Mechanics superposition does not mean interaction. Wave functions are superposed, added, on each other and the system of photons described by the sum is not interacting, ( two photon interactions are very rare at low photon energies). By the rules of quantum mechanics it is this new, superposed, wave function,when complex-conjugate squared, that is measurable in the laboratory.

Incoming photons in a beam are described by a plane wave solution of the equation, a single photon, and the superposition of the zillions of photons that make up the incoming plane wave beam, are coherent and each single photon is described by the plane wave solution.

At the level of the slit, boundary conditions enter in the solution of the quantum mechanical equation: the width of the slit, the distance between slits and whatever else is in the hardware of the experiment. The wave function is no longer a plane wave solution.

Boundary solutions generate wave functions that have a probability distribution ( i.e. the complex conjugate square of the wave function no longer displays the uniform behavior of a plane wave). Each individual photon is described by this solution and the beam, if it has kept its coherence,( which it does in the double slit set up) will be a superposition of individual photon wave functions. That is why, either single photon at a time or a coherent beam show the same interference pattern.

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 way single photons build up a coherent classical electromagnetic wave is not simple, and can be seen here, but it needs a QED background to understand it.