Hermitian Operators: Identifying & Solving Examples

[Axiom17]

Homework Statement

I have some operators, and need to figure out which ones are Hermitian (or not).

For example:

1. $$\hat{A} \psi(x) \equiv exp(ix) \psi(x)$$

Homework Equations

I have defined the Hermitian Operator:

$$A_{ab} \equiv A_{ba}^{*}$$

The Attempt at a Solution

I just don't know where to start with this

 

[Mindscrape]

One good way is to see if something is hermitian is if

$$\langle \psi | A | \psi \rangle= \langle \psi |A| \psi \rangle^*$$

if A is hermitian then the equality will hold.

 

[Axiom17]

Ok but I don't understand the calculations to do

 

[javierR]

The way you expressed the opeator A_{ab}=A*_{ba} is a matrix notation, useful when acting on a set of vectors like v_{b}. But in your problem, how does the operator act on the wavefuntion Psi(x)? Can you re-express your definition above for Hermiticity in this specific case?

 

[Axiom17]

Umm I have this:

$$\langle a|\hat{A}|b \rangle = \int dV \psi_{a}^{*}(r)r^{2}\psi_{b}(r)=[\langle b|\hat{A}|a \rangle]^{*}$$

But I don't know if that's any use

 

[Axiom17]

Still don't get this

 

[Mindscrape]

So, for example

$$\hat{A} \psi(x) \equiv exp(ix) \psi(x)$$

$$\langle \psi | A | \psi \rangle= \int \psi^*(x) exp(ix) \psi(x)$$

$$\langle \psi | A | \psi \rangle^*= \int \psi^*(x) exp(ix)^* \psi(x)=\int \psi^*(x) exp(-ix) \psi(x) \neq \int \psi^*(x) exp(ix) \psi(x)$$

The operator is not hermitian.