I need the dirac notation expectation value explaining to me please?

[jeebs]

Hi,

I find a lot of the time in QM i have been calculating things blindly. Take the expectation value for instance. I have worked this out in integral form plenty of times, but haven't really understood why I'm doing what I'm doing. I looked up wikipedia and apparently, for a measurable quantity/Hermitian operator A & a system in the state $$\psi$$, the expectation value is $$<A> = <\psi|A|\psi>$$. As I understand it, the expectation value of a measurement is the mean value if we repeated the measurement loads of times. Given this definition, it is not obvious to me why the equation above works. I don't see that just from looking at that equation, can anyone explain?

Anyway, I will take it for granted that $$<A> = <\psi|A|\psi>$$. I know that if the operator A is expressed matrix-style using its orthonormal eigenbasis $$|\phi_j>$$, then the eigenvalues a_j found from the eigenvalue equation $$A|\phi_j> = a_j|\phi_j>$$ will appear on the leading diagonal. My notes also state that for a complete orthonormal basis, any vector $$|\psi>$$ can be written as $$|\psi>=\sum_j c_j|\phi_j>$$. I think it's okay to say that in this particular basis, c_j = a_j, so that we have $$A|\psi> = A\sum_j a_j|\phi_j> = \sum_j a_jA|\phi_j> = \sum_j a_j^2|\phi_j>$$. I then stick the bra on the other side, to make $$<A> = <\psi|A|\psi> = <\psi|\sum_j a_j^2|\phi_j> = \sum_j a_j^2<\psi|\phi_j>$$.
Apparently this is wrong though, wikipedia states that $$<A> = <\psi|A|\psi> =\sum_j a_j|<\psi|\phi_j>|^2$$.

I cannot see why that is or where I have went wrong, but I get the feeling I must be confused over something really fundamental and simple. If anyone could clear this up I'd really be grateful, it would probably stop me making mistakes in other areas as well.
Thanks.

 

[gabbagabbahey]

Hi,

I know that if the operator A is expressed matrix-style using its orthonormal eigenbasis $$|\phi_j>$$, then the eigenvalues a_j found from the eigenvalue equation $$A|\phi_j> = a_j|\phi_j>$$ will appear on the leading diagonal.

That should immediately tell you that you can write

$$A=\sum_i a_i| \phi_i\rangle\langle \phi_i |$$

My notes also state that for a complete orthonormal basis, any vector $$|\psi>$$ can be written as $$|\psi>=\sum_j c_j|\phi_j>$$. I think it's okay to say that in this particular basis, c_j = a_j

Why would you think that the coefficients of your wavefunction in this eigenbasis are the same as the eigenvalues of your operator? Certainly that won't be true for every wavefunction, will it?

Apparently this is wrong though, wikipedia states that $$<A> = <\psi|A|\psi> =\sum_j a_j|<\psi|\phi_j>|^2$$.

I cannot see why that is

Simple,

$$\begin{aligned}\langle \psi|A|\psi\rangle &= \langle \psi|\left(\sum_i a_i| \phi_i\rangle\langle \phi_i |\right)|\psi\rangle \\ &= \sum_i a_i\langle \psi| \phi_i\rangle\langle \phi_i |\psi\rangle \\ &= \sum_i a_i\left|\langle \psi| \phi_i\rangle\right|^2\end{aligned}$$

 

[jeebs]

thanks man

Why would you think that the coefficients of your wavefunction in this eigenbasis are the same as the eigenvalues of your operator? Certainly that won't be true for every wavefunction, will it?

this is exactly the sort of daft mistake I need to iron out.