Symmetric and Asymmetric Waves
When describing objects from the macroworld, we often use words like “identical” or “same”. We could proclaim, for instance, that two mobile phones of the same model are the same. The problem is, however, that no two objects from the macroworld are actually “the same”. There is at least a slight difference between any two objects from the macroworld. With the mobile phones mentioned earlier, the difference is not visible at first sight, since it is on a molecular level. In addition, one can always simply differentiate between the phones by marking them (for example, one can paint one of the phones blues and the other one red).
In the microworld, however, the words “identical” or “indistinguishable” have a completely different meaning. Any two electrons (protons, photons, etc.) are absolutely identical and there is no way of telling them apart. One cannot mark them in order to make them different either (it is simply not possible to “paint” an electron, since color has no meaning in the microworld). It therefore makes no sense to refer to two electrons as “the first electron” and “the second electron”, since there is no way of telling which, one is which.
Let us consider two identical particles, one of them is described by the wave function ψ (1), the other is described by the wave function ψ (2):
We could describe these two particles by a combined wave function that is a combination of the two original wave functions ψ (1) and ψ(2). This wave function would take a form ψ (1,2) = ψ (1)ψ(2):
But what happens if we swap the particles (i.e., the particle that was originally assigned the wave function ψ (1) is now assigned the wave function ψ (2) and vice-versa) and describe them by a combined wave function in the form of ψ (2,1) = ψ(1)ψ(2)? The particles are indistinguishable, so we should not be able to spot any difference after we swap the particles, and the system should look exactly the same as before the particles were swapped. One can achieve that only if the wave function ψ (1,2), which describes the system before the particles are swapped, is identical to the wave function ψ (2,1), which describes the system after we swap the particles, therefore:
𝛙(𝟏,𝟐) = ±𝛙(𝟐,𝟏)
In some cases, however, it may happen that the wave function changes its sign after the particles are swapped. In case this happens, the wave function is considered to be antisymmetric. If the sign remains preserved, it is a symmetric wave function. Bosons are described by symmetric wave functions; antisymmetric wave functions are typical for fermions.