- #1
spookyfw
- 25
- 0
Hi Folks,
I somehow cannot get the difference and have to admit that I am left confused.
For a probability of measuring m with the operator M on state [itex]\Psi_i[/itex]
[itex]p(m|i) = <\Psi_i| M^{+}_m M_m |\Psi_i> = <\Psi_i| M_m |\Psi_i>[/itex].
The average of an observable is defined as [itex] <O> = <\Psi_i| O |\Psi_i>[/itex].
So the measurement by M gives me a probability, the measurement with the observable an expectation value? Okay..the observables will be hermitean, the only thing I know about the measurement matrix is, that is not unitary - otherwise [itex] M^{+}_m M_m[/itex] would be equal to the unity matrix.
Is the difference that an observable doesn't change the system, but a measurement when projective projects the system into one of the states?
One last question: What is the reasoning for defining a measurement like [itex] M^{+}_m M_m [/itex] and not by M alone directly?
Thank you so much in advance..I hope the above somehow makes sense ;).
Steffen
I somehow cannot get the difference and have to admit that I am left confused.
For a probability of measuring m with the operator M on state [itex]\Psi_i[/itex]
[itex]p(m|i) = <\Psi_i| M^{+}_m M_m |\Psi_i> = <\Psi_i| M_m |\Psi_i>[/itex].
The average of an observable is defined as [itex] <O> = <\Psi_i| O |\Psi_i>[/itex].
So the measurement by M gives me a probability, the measurement with the observable an expectation value? Okay..the observables will be hermitean, the only thing I know about the measurement matrix is, that is not unitary - otherwise [itex] M^{+}_m M_m[/itex] would be equal to the unity matrix.
Is the difference that an observable doesn't change the system, but a measurement when projective projects the system into one of the states?
One last question: What is the reasoning for defining a measurement like [itex] M^{+}_m M_m [/itex] and not by M alone directly?
Thank you so much in advance..I hope the above somehow makes sense ;).
Steffen