Does the Parity Operator Commute with All Hermitian Operators?

[roshan2004]

In my terminal exam I was asked to prove it that the parity operation commutes with Hermitian operator? I wonder how can we show that? coz we can only show that the parity operator is hermitian? We haven't got the value of hermitian operator at all?

 

[Petr Mugver]

On wave functions the parity operator acts something like

$$P\psi(x)=\eta\psi(-x)$$

where \eta is a phase factor independent of x. Now it should be easy to prove all the commutation relations you need.

 

[roshan2004]

Pls Can you show it to me how?

 

[Petr Mugver]

I can try, but first you have to tell me which is the operator you want to calculate the commutator with parity.

 

[roshan2004]

Hermitian operator

 

[Petr Mugver]

Hermitian operator

It's not enough. For example, P does not commute with the coordinates x or the momenta p (these anticommute with parity), but it commutes with orbital agular momentum or spin...

 

[roshan2004]

So, wasn't the question a wrong one?

 

[the_house]

If it really implied that the parity operator commutes with all Hermitian operators, then yes, it was wrong.