Materials in Electronics/Confined Particles/Parity

Parity in 1 Dimension

Parity is one dimension is essentially a statement of the wavefunction's "evenness" or "oddness". This is how the wavefunction behaves when it is reflected about the point x=0:

$\psi \left(x\right)=\psi \left(-x\right)$		$\psi \left(x\right)=-\psi \left(-x\right)$
Even Parity		Odd Parity

Parity is important in scenarios such as the 1D finite square potential well. Because the well is symmetrical about its midpoint, it follows that the electron's probability density is also symmetrical about the midpoint (i.e. even). Recall that the probability density is given by the wavefunction multiplied by its conjugate. The statement of the evenness of the porobability is therefore:

$\psi \left(x\right)^{*}\psi \left(x\right)=\psi \left(-x\right)^{*}\psi \left(-x\right)$

The theory of even and odd functions shows that either an odd or even function multiplied by itself yields an even function, but a linear compination will not. Therefore, this condition can only be fulfilled if the wavefunction parity is either even or odd. It cannot be fulfilled if the wavefunction parity is a linear combination of even and odd functions, as this kind of combination produces neither an even or odd function.

Generally speaking, any wavefunction must have a definite parity with respect to the symmetry of the physical system