Show that the eigenvalues of a hermitian operator are real.

[leoflindall]

Homework Statement

Show that the eigenvalues of a hermitian operator are real. Show the expectation value of the hamiltonian is real.

Homework Equations

The Attempt at a Solution

How do i approach this question? I can show that the operator is hermitian by showing that T_mn = (T_nm)* with no problems.

I know that the outcome of a measurement must be real, so;

<Q> = <Q>*

Do I need to apply a Hermitian operator to a wave function, and determine the expectation value and show that this satisfys the above condition?

And if so how do i show this in general?

*****UPDATED*****
I have found the following proof (Intro to quantum mechanics, griffiths)

Suppose Q^{^} f = q f, (1)

(f(x) is an eigenfunction of Q^{^} , with eigenvlaue q), and;

< f l Q^{^} f > = < Q^{^} f l f > (2)

then

q < f l f > = q* < f l f > (3)

as < f l f > cannot be zero, then q must equal q*, and thus the eigen values are real.

Is it possible to do this proof in intergral form? I kind of understand this, but any additional explanation of the step between (2) and (3) would be really helpfull.

How do I follow on from this to show that the expectation value must also be real?

I know this isn't hard but have managed to confuse myself, and advice would be greatly appreciated.

Leo

 

[Kaleb]

Remind yourself what the integral form of expectation value is. Write it down, then remember what was taught in your class about these inner products. :) Should be simple from there.

 

[dextercioby]

Do you have somewhere in your notes a definition for the expectation value of an observable represented by an operator A in an arbitrary state represented by a unit vector $\psi$ ? Use it and use the fact that, if A can be chosen for simplicity as only posessing a spectrum made up only of eigenvalues and A is self-adjoint, the Hilbert space A <lives> in admits a denumerable basis made up only from A's eigenvectors. On a second thought, trash that. It's an unnecessary complication. Just use the definition of expectation value and the one for hermiticity (or symmetry for the mathematically minded reader).