ðŸŒ¿ A short note on symmetries in Quantum Mechanics

In Quantum Mechanics, when we consider symmetries, we look at the Hamilton operator, and see if it is constant under some transformation of another operator.

For example, consider the Hamilton for the Quantum Harmonic Oscilator, then $V(x)=V(-x)$ has parity. That is

$$\hat{\pi} \hat{H}(x,p) f(x) = \hat{H}(-x,-p) f(-x) = \hat{H}(x,p) \hat{\pi} f(x)$$

This means that $[H,\pi]=0$ $\Rightarrow$We can choose common eigenfunctions for $\hat{H},\hat{\pi}$.

If we choose the common eigenfunctions, the eigenfunctions of $\hat{H}$ must follow the properties of the eigenfunctions for $\hat{\pi}$. There are two kinds of eigenfunctions for $\hat{\pi}$.

- Even functions: eg. $\hat{\pi}\cos(x)=\cos(-x)=\cos(x)$ with eigenvalue $1$

- Odd functions: eg. $\hat{\pi} x = -x$ with eigenvalue $-1$.

So we know that the eigenfunctions for $\hat{H}$ must be either even or odd.

From the properties of $\hat{\pi}$, we can construct selection rules to quickly determine expectations, and evaluate various integrals.

In general

$[\hat{H},\hat{Q}]=0$ only when there is a symmetry, and per Ehrenfest's generalized theorem, we know that $\hat{Q}$ does not change in time, so it's conserved. Finally, we consider the eigenvalues of $\hat{Q}$ good quantum numbers because they do not vary in time. This is all summarized in Noether's theorem.