Proving a squared Hermitian operator is Hermitian

[kehler]

Homework Statement

Homework Equations

The Attempt at a Solution

I've gone round in circles doing this! I started of by writing it as an integral of (psi* x A_hat² x psi) w.r.t dx, then using the equation above but I keep coming back at my original equation after flipping it around. Are there any tricks in particular that I should know when dealing with squares of operators? Or any assumptions that I need to make? Please help :(

 

[turin]

It may help to insert a "resolution of the identity" (as Shankar calls it). The operators can be treated in terms of their "matrix elements" on the Hilbert space, which is the space of the functions, Ψ, on which the operators act.

Are you familiar with the Dirac (bra-ket) notation? I would start with Dirac notation, and then simply insert the appropriate identity resolutions.

 

[kehler]

What's a resolution of the identity? I don't have the Shankar text. We use Griffiths in class... No unfortunately I'm not too familiar with Dirac Notation :(. I know roughly what it is but I'm not confident in using it just yet. So far we've just been taught the integral form. I think we're just meant to manipulate the equation till we get to be the required function but I've tried that several times without getting anywhere

 

[turin]

OK. The first thing to do is to express the expectation value explicitly. Then, split A² into factors of A. Then, use your definition for Hermiticity in a clever way.

 

[kehler]

I've done the first two steps. It's the third that I'm having trouble with.

What do I do from here? :s

 

[turin]

You didn't really do the second step, then. Think of the action of each A independently. That is, think of the action of the first A on the wavefunction as producing a new wavefunction, and then the other A acts on this new wavefunction.

 

[kehler]

Ahh ok I see :). Thanks. Gosh it seems really easy now.