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Create Symmetric and Antisymmetric Wave Functions for Any System of N Particles

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,

image0.png

To create a symmetric wave function, add together

image1.png

and the version where the two particles are swapped,

image2.png

Assuming that

image3.png

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

image4.png

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

image5.png

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

image6.png

look like? Why, it’d look like this:

image7.png

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

image8.png

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

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