**Required math: calculus**

**Required physics: 3-d Schrödinger equation**

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 5.4.

One of the more unusual postulates of quantum mechanics is that particles of the same type (for example, 2 electrons) are actually identical, in the sense that we cannot distinguish one from the other by any means at all. This is a radical departure from classical mechanics, where all particles are, in principle, different, even if they have the same physical properties. In classical mechanics, it makes sense to talk about ‘electron #1′ and ‘electron #2′, but in quantum mechanics, all electrons (or particles of other types such as protons and neutrons) are exactly the same, and it is impossible to label them in any way.

Up to now, we’ve been considering mainly one-particle systems moving in potentials, so the issue of identifying one of a group of particles hasn’t arisen. We’ve essentially assumed that the one particle (usually an electron) is the only particle in the universe, so we don’t need to think about labelling it.

For two (or more) particles, though, we do need to think about how to describe the quantum state in such a way that, even from the mathematics, we can’t tell which particle is which. For a single particle moving in a potential, we can solve the time-independent Schrödinger equation and get a set of stationary states. Once we’ve done that, we can generate the time-dependent solution in the usual manner by constructing a series:

If we have 2 particles, we need to introduce two spatial coordinates, one for each particle. Now suppose that each particle is in one of the stationary states, say and . If the particles are really indistinguishable, then it shouldn’t be possible to tell which particle is in which state. That is, trying a total wave function of the form won’t work, because we’ve associated particle with state and with state .

One way of mixing things up is to take a symmetric or anti-symmetric combination of the two states. That is

where is a normalization constant. Assuming the individual functions are orthonormal, we can work out . Using the orthonormal properties of the functions, we get, assuming :

so .

In the case where , the minus sign results in the total wave function being zero, so we have only the case with the plus sign. In this case, we get

so here .

Particles for which we take the plus sign above are known as *bosons*, and particles using the minus sign are known as *fermions*.

## Trackbacks

[...] the particles are bosons, then they are identical and the wave function is a symmetric sum, so we [...]

[...] seen that distinguishable particles and identical particles must be treated differently in quantum mechanics, resulting in different combinations of the [...]

[...] extension of the two-particle wave functions for identical particles to higher numbers of particles is relatively straightforward. The condition [...]

[...] far, we’ve looked at identical particles only in the non-interacting case. In real life, of course, most particles interact with each other, [...]

[…] far, we’ve looked at identical particles only in the non-interacting case. In real life, of course, most particles interact with each other, […]

[…] example of perturbation theory applied to the interaction between two particles. We place two bosons in an infinite square well, where the single particle wave functions are given […]