Check Answers to Quantum Minima Qs: In-State Vector Operator

[Beer-monster]

I'm trying to "sure"up my Quantum Mechanics and found an sheet of mostly conceptual review questions. A sort of "Quantum minima".

I'd like to check my answers to a couple of these.

Firstly:

In terms of state vector the wavefunction is the expansion coefficient (probability amplitude) of a state i.e

\psi = \langle x \left | \psi \right\rangle

What happens if you put an operator (e.g. p) between the bra and ket as if you were calculating an expectation value? i.e.

\left\langle x \left | p \right | \psi \right \rangle

My guess is that the action of the operator on the state vector \left|\psi\right\rangle will collapse the vector to a single basis vector of p \left| p \right\rangle the expression above would reduce to an element of a basis transformation matrix \left\langle x | p \right\rangle

Is that correct...at least in part?

 

[dextercioby]

Not quite. The action of p on the ket psi leads to a ket phi, generally different. Then the matrix element with the bra x leads to a complex number, namely phi(x).

 

[Beer-monster]

Thanks for the reply.

What would the physical interpretation of the complex number phi(x) be? The wavefunction of the state with eigenvalue of p?

 

[dextercioby]

No, it's a generic wavefunction. No relation to spectral values of p.