# Determine whether or not is a Hermitian operator

1. Dec 5, 2011

### dje

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

The operator F is defined by Fψ(x)=ψ(x+a) + ψ(x-a), where a is a nonzero constant. Determine whether or not F is a Hermitian operator.

2. Relevant equations

∫(x+a)d/dx + (x-a)d/dxψ

3. The attempt at a solution

f = (1=ax) + (1-ax)ψ

What are the steps I need to do to figure this out. Thanks.

2. Dec 6, 2011

### dextercioby

No matter the value of a, one can show that F is bounded, so the adjoint of it exists. Then all you need is to check is the hermiticity condition in integral form:

$$\int dx \psi^{*}(x) F\psi(x) = ?$$

Try to get the psi with exchanged argument under the complex conjugate sign.

3. Dec 6, 2011

### dje

I am not sure of these steps but I will try. Can you show me if I am still not understanding this. thanks.

Fψ(x)=Fψ(x+a) + ψ(x-a) Fτ= F to be Hermitian
Fψ (x+a) + (x-a) = F dt/dx? (x+a) + (x-a)

= F dt/dx (x + a) + (x-a)

∫(x+a)d/dx + (x-a)d/dx ψ

F τ= (1+ax)ψ + (1-ax)ψ

KEY= * below/symbol I am wanting here is circle with vertical line through it.
(θ*/ψ) = (ψ/θ*)
θ* (x) ψ(x+a) + ψ(x-a) dt/dx dx
= ψ(x +a) + ψ (x-a) dθ*/dx dx

Solution- F is Hermitian operator Fτ= F