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leoflindall
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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 Tmn = (Tnm)* 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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