What logical/mathematical proof do we have to show that, everything which is called electron are identically the same?

Question

Premise 1: Physics don't believe in sense "organs" of the human "robot" (more commonly said "common sense deceives us").

Premise 2: Physics believes in logic or mathematics.
Background thrust: Quantum mechanics.

Premise 3: Everything which "revolves" around the nucleus might not have identical properties. There might be some property of particles (revolving around the nucleus) which are not the same, because of our sense not detecting it. Then, we might say the particles to be identical only w.r.t our senses, but that is not the spirit of our physics, it must be proved logically. Or else we might define particles of such and such properties to be such and such, but that doesn't define them to be entirely identical.

So, are particles revolving around the nucleus identical in all properties in reality? Or is there any logical or mathematical proof to show that the particles are all identical?

The same argument can be applied to all the particles which we call identical.

Answer 1

In physics (as in all natural sciences) there are never any proofs. We write theories and believe them as long as there are no experimental results contradicting them. You can prove that a theory is wrong, but we can never be sure that it is correct.

However, your question can be answered in a convincing way by looking at Fermi-Dirac and Bose-Einstein statistics. It is a property of quantum physics that exchanging the position (and state) of two identical particles can only change the wave function by multiplying it by $\pm 1$, $$\psi(x_1,x_2) = \pm \, \psi(x_2,x_1) \, .$$ The square of the $\psi(x_1,x_2)$ is the probability to find the particle at the positions $(x_1,x_2)$. When the $+1$ applies we call the particles bosons and when it's $-1$ we have fermions. It turns out that electrons are fermions.

Now imagine that you have two identical fermions at the same position in space. Then we have $$\psi(x_1,x_1) = - \, \psi(x_1,x_1) \, ,$$ and we conclude that $\psi(x_1,x_1) = 0$. This means that two identical fermions can not be in the same quantum state. They do not overlap. Note that we are talking about non-interacting particles here. There is no repulsion pushing them apart.

Simple quantum mechanics of two charged interacting particles provides us with a detailed structure of the quantum states that exist around the nucleus of an atom for its electrons. Then fermi-dirac statistics can be applied by assuming that multiple electrons can be bound to a nucleus without interacting with each other. We find that all the lowest energy states are occupied by the available electrons. More importantly, the number of different occupied states is the same than the number of electrons. This is not a very good way to minimise the energy of the atom. If the electrons could all occupy the same quantum state (if they were different), they would all fall down to the lowest energy state.

In this sense, atoms would not be stable and the periodic table of elements would not have the structure that is has is electrons were not indistinguishable fermions.

Note that for Bosons, all particles do tend to occupy the lowest energy state. This phenomena is called Bose-Einstein condensation and is made to happen in labs all around the world.

Answer 2

The most fundamental description of the electron we have at the moment is using quantum field theory. This describes the electron as an excitation of a single quantum field that spans all of time and space. It also neatly explains how electrons can be created and destroyed: add a quantum of energy to the quantum field and it appears as a newly created electron. Remove a quantum of energy from the field and an electron disappears.

Quantum field theory is the best tested theory we have, so even though it may seem an odd way of looking at the world we have a lot of confidence that it provides a good description of the physical world around us.

And it also neatly explains why electrons are all identical - they are all identical because all electrons are quanta of the same quantum field. The same applies to all the fundamental particles, so all up quarks are identical because they are all quanta of the same up quantum field, all Z bosons are identical because they are all quanta of the same Z quantum field, and so on.

Answer 3

As I understand it, to be able to distinguish between two things, is to be able, by some comparison between them , to find a property that is different between them. In macroscopical classical physics we are able to say a body 1 to be different than a body 2, even if they are identical in shape and all properties of their form because, if we have them in front of us we can say the body 1 is at position x1 and to to x2, in our coordinates system.

For particles, that is for quantum mechanic "bodies", there isn't such an ability.I' m quoting from Wikipedia(http://en.wikipedia.org/wiki/Identical_particles) (which has some issues, but for our subject they are irrelevant)

" There are two ways in which one might distinguish between particles. The first method relies on differences in the particles' intrinsic physical properties, such as mass, electric charge, and spin. If differences exist, we can distinguish between the particles by measuring the relevant properties. However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties. For instance, every electron in the universe has exactly the same electric charge; this is why we can speak of such a thing as "the charge of the electron". Even if the particles have equivalent physical properties, there remains a second method for distinguishing between particles, which is to track the trajectory of each particle. As long as one can measure the position of each particle with infinite precision (even when the particles collide), then there would be no ambiguity about which particle is which. The problem with the second approach is that it contradicts the principles of quantum mechanics. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by wavefunctions that give the probability of finding a particle at each position. As time passes, the wavefunctions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles are then said to be indistinguishable.

Consider two electrons in a head-on collision. Say you have an electron with spin up approaching from the left, and an electron with spin down approaching from the right. Then after the collision you have an electron with spin up leaving to the left, and an electron with spin down leaving to the right. You could say "The electron with spin up bounced off and went back the way it came, and so did the one with spin down." Or you could say "The two electrons passed right by each other, but they each flipped their spins on the way." But the point is that in quantum mechanics you can't say which of these processes "really happened", because there's no difference between the two. Of course, this is pretending the electrons have a definite position and velocity, when in fact the probability of finding them to have any particular position or velocity is part of the state. So really you might have started with "an electron in spin up that's highly likely to be approaching from the left", and "an electron in spin down that's highly likely to be approaching from the right" -- but there's a small probability of finding the spin up electron over by where the spin down one was, or vice versa. (Of course you can also find them somewhere else altogether.) Another way to say this: We have two terms in our antisymmetric wave function, but when the electrons are far apart then for a certain pair of positions (call them x1 and x2) the amplitude of one term (spin up at x1 and spin down at x2) is much greater than the amplitude of the other term (spin up at x2 and spin down at x1). However, as the electrons get near each other, the probability of finding the spin up electron where the spin down electron used to be, or vice versa, becomes as great as the probability of finding them where you thought they were. In other words, the amplitudes of the two terms in our antisymmetric function (for whatever x1 and x2 we want to look at) both contribute significantly. At that point, we can't say "we have the spin up particle over here", and "we have the spin down particle over there", because we're just as likely to measure it the other way around. This can be understood as a consequence of the uncertainty principle. Hopefully you can see what I'm getting at. If we have two electrons in two different states, such as spin up and spin down, we can talk about the electron with spin up and the electron with spin down. But as they get close together, you're just as likely to find the spin up one where you thought the spin down one was, or vice versa -- so how do you know it's the same electron in a given spin state from one moment to the next? You don't, really. So while you can talk about "the electron in state 1" and "the electron in state 2", really you just have two electrons and two internal states distributed among them. "

I hope this helps. The wave functions, down at the quantum level are the reason we can't distinguish the trajectories of the particles(and the wave functions give the possibility of finding the particle-if two electrons are close we cannot speak of indistinguishable trajectories)