Quantum mechanics expectation of an operator

[mtp !EKFQOBEsKo No.6340665]

say ive got an operator A that is a 2x2 matrix with eigenvalues 1 and -1 and some wave function psi. does the expectation value of <A>=1 or -1 or can it equal something else

i was pretty sure i saw somewhere that it has to be an eigenvalue but cant find it anywhere and an answer i got to a question was between -1 and 1. also googling that shit is not easy

[Anonymous No.6340712]

Observables are eigenfunctions of the operator.
The expectation value can be any real number, or in some cases, infinite.
So, once the system is measured, it will collapse to an eigenstate. (In reality, it only collapses to a narrow spike around an eigenvalue). The eigenvalue is the measurement.
In this case, with eigenvalues 1 and -1, the expectation value will be somewhere between 1 and -1. This is because observable operators are hermetian, so their eigenfunctions form an orthogonal basis, so any possible wavefunction is a superposition of eigenfunctions with eigenvalue -1 or 1.

[mtp !EKFQOBEsKo No.6340739]

>>6340712
okay thanks that makes sense, unlike than what i thought i read

[Anonymous No.6340755]

you're thinking that you 'saw somewhere' is correct. It can indeed only be 1 or -1 if those are the eigenvalues.

The possible outcomes of a measurement of an observable are quantized to the eigenvalues of the associated matrix.

[Anonymous No.6340764]

>>6340755
OP is talking about expectation values, not outcomes of a measurement.