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

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

**2. Relevant equations**

**3. 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

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