Measurement values affects probabilities in QM?

Question

Consider a non-degenerate operator $\Omega$ with discrete eigenvalues $\omega_i$, where $i=1,2,3,...$. We can write $\Omega = \sum_i \omega_i~|\omega_i\rangle \langle \omega_i|$, where $|\omega_i\rangle$ are the normalized eigenvectors. Any arbitrary state may then be expanded as: $|\psi\rangle = \sum_j c_j~|\omega_j\rangle$, where $|c_j|^2$ gives the probability that the state will collapse to $|\omega_j\rangle$ if the measurement corresponding to $\Omega$ were to be performed.

Also the action of $\Omega$ on a state vector yields a new state vector: $$|\psi '\rangle = \Omega |\psi\rangle = \sum_{i,j}\omega_ic_j~|\omega_i\rangle \langle \omega_i|\omega_j\rangle = \sum_{i,j}\omega_ic_j~|\omega_i\rangle \delta_{ij} = \sum_{i}\omega_ic_i~|\omega_i\rangle.$$ The probability that this new state will collapse to $|\omega_i\rangle$, if the measurement $\Omega$ were to be performed, is proportional to $|\omega_ic_i|^2$. The probabilities have now become a function of the possible values for the operator $\Omega$. If operated on by some other operator $\Lambda$ then the probabilities will become a function of its eigenvalues $\lambda_i$, and so on.

I have two questions:

(1) Is my conclusion derived above correct?

(2) What is the difference between an operator that changes the state of a system in general and an operator that corresponds to a measurement? Of course, a measurement also changes the state of system but this change is specific: A measurement always transforms a state vector into one of its eigenstates. This can't be represented mathematically (correct me if I am wrong) because $\Omega |\psi\rangle\neq$ an eigentstate of $\Omega$ (in general).

Answer 1

Consider a non-degenerate operator $\Omega$ with discrete eigenvalues $\omega_i$, where $i=1,2,3,...$. We can write $\Omega = \sum_i \omega_i~|\omega_i\rangle \langle \omega_i|$, where $|\omega_i\rangle$ are the normalized eigenvectors. Any arbitrary state may then be expanded as: $|\psi\rangle = \sum_j c_j~|\omega_j\rangle$,

It has not been said explicitly in the question, but I will assume that your $\Omega$ is an observable. This means that $\Omega$ is a Hermitian operator whose eigenvectors span the space. Therefore, as you posit above, an arbitrary state can be expanded in its eigenvectors. This may not be the most general case, but I think it is sufficiently general for our purposes.

where $|c_j|^2$ gives the probability that the state will collapse to $|\omega_j\rangle$ if the measurement corresponding to $\Omega$ were to be performed.

Also the action of $\Omega$ on a state vector yields a new state vector: $|\psi '\rangle = \Omega |\psi\rangle = \sum_{i,j}\omega_ic_j~|\omega_i\rangle \langle \omega_i|\omega_j\rangle = \sum_{i,j}\omega_ic_j~|\omega_i\rangle \delta_{ij} = \sum_{i}\omega_ic_i~|\omega_i\rangle$. The probability that this new state will collapse to $|\omega_i\rangle$, if the measurement $\Omega$ were to be performed, is proportional to $|\omega_ic_i|^2$. The probabilities have now become a function of the possible values for the operator $\Omega$. If operated on by some other operator $\Lambda$ then the probabilities will become a function of its eigenvalues $\lambda_i$, and so on.

Well, sure, but so what? Why would you act with just $\Omega$? Remember that the evolution of quantum systems is unitary and $\Omega$ is not necessarily unitary. For example, if $\Omega = H$ then the system evolves in time by the action of the unitary operator $e^{-i\Omega t}$, not by the action of the Hermitian $\Omega$ alone. As another example, if $\Omega = S_z$ then the system rotates about the z-axis by the action of the unitary operator $e^{-i\theta \Omega}$, not by the action of the Hermitian $\Omega$ alone.

If you are not trying to evolve the system, but instead are interested in, say, the expectation value of the measurement of $\Omega$ then you would be interested in $\Omega|\psi\rangle$ in order to calculate $\langle\psi|\Omega|\psi\rangle=\sum_j |c_j|^2 \omega_j$, which (surprise, surprise) is just the sum of the possible measurement values weighted by the probability that the value is measured.

I have two questions:

(1) Is my conclusion derived above correct?

Yes.

(2) What is the difference between an operator that changes the state of a system in general and an operator that corresponds to a measurement?

This has already been explained above, but I will recap here. Measurements correspond to real values, which are the eigenvalues of observable operators. When you measure you obtain one of the eigenvalues and the probability that you obtain a given value is $|c_j|^2$, in our notation. See also this link that goes into more detail about observables, projective measurement, and other measurement operators.

Evolution of the state (other than collapse) is performed by the action of unitary operators, which do not generally correspond to measurements since they are unitary and not Hermitian. See also this link regarding unitarity.