Understanding Expectation Values in Quantum Mechanics

[dudy]

Let A be an observable (opeator), and we're assuming that for a given state psi(x), the value of A is given by A acting on psi(x), namely - A|psi>.
Also we assume that - P(x) = |psi(x)|^2
So, I'de expect the Expectation value of A to be defined like so:
<A> = Integral[-Inf:+Inf]{ P(x) A psi(x) dx} = Integral[-Inf:+Inf]{ |psi(x)|^2 A psi(x) dx} , which is not <psi|A|psi>, and that's clearly not right. where did i go wrong here?

 

[jtbell]

Let A be an observable (opeator), and we're assuming that for a given state psi(x), the value of A is given by A acting on psi(x), namely - A|psi>.

No. A|\psi\rangle is not a value. It's another state, in general.

If |\psi\rangle happens to be an eigenstate of the operator A, then A|\psi\rangle = a|\psi\rangle, where a is an eigenvalue of the operator A.

 

[dudy]

got it, thank you!