Quantum operator hermiticity. Show that S is Hermitian

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>

How should i go on about this?

1. The problem statement, all variables and given/known data







2. Relevant equations







3. The attempt at a solution