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Nickweynmann
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Homework Statement
Using Dirac Notation prove for the Hermitian operator B acting on a state vector |ψ>, which represents a bound particle in a 1-d potential well - that the expectation value is <C^2> = <Cψ|Cψ>.
Include each step in your reasoning. Finally use the result to show the expectation value of the kinetic energy of the particle in the state |ψ> is <Ekin> = (1/2m)<Pψ|Pψ> ?
Can someone enlighten my thought via a diagram exactly what I am looking at pls
Homework Equations
I have to find the right eqns, which I guess are the Hermitian related eqns: stating what is Hermitian
<a|Cb> = <Ca|b> , where C is operator, a and b are normalisable functions.
<a|λCb> = λ<a|Cb> , where λ is a real constant.
any power of C is Hermitian
The Attempt at a Solution
My attempt has been quite measely as I cannot get any visual sense of it. (ideas?)
So> To prove the <C^2> = <Cψ|Cψ>, all I can think to say is...
The RHS with C being Hermitian.
<Cψ|Cψ> = <ψ|CCψ> = <ψ|C^2ψ>
As we are dealing with the same operator there is no difference if the order in which they so therefore obeys the requirement for a product to commute CC - CC = 0.
therfore that will give the expectation value of <C^2> = <Cψ|Cψ>.
Is that it or is there more reasoning?
Turning to <Ekin> I have no idea what to do because it seems like the (1/2m) is just stuck in there because it previously known. Is there another reasoning?