# Quantum operator hermiticity. Show that S is Hermitian

1. Apr 30, 2014

### engstruggle

1. The problem statement, all variables and given/known data
Spin Operator S has eigenvectors |R> and |L>,
S|R> = |R>
S|L> =-|L>

eigenvectors are orthonormal

2. Relevant equations
Operator A is Hermitian if <ψ|A|Θ> = <Θ|A|ψ>*

3. The attempt at a solution
<ψ|S|L> = <L|S|ψ>* // Has to be true if S is Hermitian
LHS: <ψ|S|L> = <ψ|-|L>
<ψ|-|L>* = <L|-|ψ>

Question: how do i know how S acts on any function like |ψ> ?
Could somebody provide an algorithm to find if an operator is Hermitian.
I have another example of operator P, where P|R> = |L>
P|L> = |R>
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 30, 2014

### unscientific

Use the definition of a hermitian operator:

$$\langle\psi|S|\psi\rangle^{\dagger} = \langle\psi|S^{*}|\psi\rangle = \langle\psi|S|\psi\rangle$$