Parity

Let us review the concept of parity. Parity is a consequence of space inversion.

${\begin{array}{rcl}x&\longrightarrow &-x\\y&\longrightarrow &-y\\z&\longrightarrow &-z\\&or&{\vec {r}}\longrightarrow -{\vec {r}}\end{array}}$

We propose an operator that (in the spirit of the rotation operator introduced earlier) takes an initial ket and returns a ket with the above inversion operation performed.

$|\alpha \rangle \longrightarrow \pi |\alpha \rangle$

We require that this operator in unitary and that it has the following key property (or, perhaps more precisely we define the operator through)

$\pi ^{\dagger }{\vec {x}}\pi =-{\vec {x}}$

which implies

${\vec {x}}\pi |{\vec {x}}_{1}\rangle =-\pi {\vec {x}}|{\vec {x}}_{1}\rangle =-\pi {\vec {x}}_{1}|{\vec {x}}_{1}\rangle =-{\vec {x}}_{1}\pi |{\vec {x}}_{1}\rangle$

This shows that $\pi |x^{\prime }\rangle$ is an eigenket of $x$ with eigenvalue of $-x^{\prime }$. Finally, the eigenvalues of $\pi$ are $\pm 1$ and

$\pi ^{-1}=\pi =\pi ^{\dagger }$

Position is "odd" under space inversion or "odd under the parity operator". Angular momentum, on the other hand is even.

$\pi ^{\dagger }{\vec {L}}\pi =\pi ^{\dagger }{{\vec {x}}\times {\vec {p}}}\pi ={\vec {x}}\times {\vec {p}}={\vec {L}}$

Because of this property position and momentum are called vectors or polar vectors and angular momentum is called an axial or psuedo vector. What about wavefunction? What does the parity operator do to wavefunctions? Well it depends on the wavefunction. For example, consider the spherical harmonics (the angular part of the hydrogen atom eigenstates). Some of the wavefunctions are odd under parity and some are even. (In one dimension a cosine wave is "even" whereas a sine wave is "odd".)

$\pi |Y_{l}^{m}\rangle =(-1)^{l}|Y_{l}^{m}\rangle$

Now, consider the case where a state is an energy eigenket and the parity operator commutes with Hamiltonian. Such a ket is not necessarily an eigenket of the parity operator. Consider, for example, the case of the hydrogen atom for $n=2$. Neglecting higher order pertubations to the hamiltonian, $|n=2\rangle$ can be made up of a combination of two eigenkets with different parities,

$|n=2\rangle =a_{1}|n=2,l=0,m\rangle +a_{2}|n=2,l=1,m^{\prime }\rangle$

Without any degeneracies eigenstates of the hamiltonian are indeed eigenstates of the parity operator if the hamiltonian and $\pi$ commute. This idea of parity gives rise to what is called a selection rule. Selection rules, in general, are nothing more than the statement that certain operators connect certain states ($\langle i|A|j\rangle \neq 0$ for certain $i,j$) and do not connect other states (that is, $\langle i^{\prime }|A|j^{\prime }\rangle =0$ for certain $i^{\prime },j^{\prime }$). Consider, for example, the ${\vec {x}}$ operator and two different parity eigenstates,

$\pi |\alpha \rangle =p_{\alpha }|\alpha \rangle ~~~\pi |\beta \rangle =p_{\beta }|\beta \rangle ~~~p_{\alpha ,\beta }=\pm 1$

then

$\langle \beta |{\vec {x}}|\alpha \rangle =0~{\rm {unless}}~p_{\alpha }=-p_{\beta }$

One can see this in the following way

$\langle \beta |{\vec {x}}|\alpha \rangle =\langle \beta |\pi ^{-1}\pi {\vec {x}}\pi ^{-1}\pi |\alpha \rangle =p_{\alpha }p_{\beta }(-1)\langle \beta |{\vec {x}}|\alpha \rangle$

which can be true only if $p_{\alpha }p_{\beta }=-1$. ${\vec {x}}$ is an "parity odd operator" and it connects states of opposite parity. "Even operators" connect states of the same parity.