In quantum physics, many of the wave functions that are solutions to physical setups like the square well aren’t inherently symmetric or antisymmetric; they’re simply asymmetric. In other words, they have no definite symmetry. So how do you end up with symmetric or antisymmetric wave functions?

The answer is that you have to create them yourself, and you do that by adding together asymmetric wave functions. For example, say that you have an asymmetric wave function of two particles,

To create a symmetric wave function, add together

and the version where the two particles are swapped,

Assuming that

are normalized, you can create a symmetric wave function using these two wave functions this way — just by adding the wave functions:

You can make an antisymmetric wave function by subtracting the two wave functions:

This process gets rapidly more complex the more particles you add, however, because you have to interchange all the particles. For example, what would a symmetric wave function based on the asymmetric three-particle wave function

look like? Why, it’d look like this:

And how about the antisymmetric wave function? That looks like this:

And in this way, at least theoretically, you can create symmetric and antisymmetric wave functions for any system of N particles.