In the ideal measurement model, what possible results can be obtained when measuring an observable \(\hat A\)?

State the Born rule for a state \(|\psi\rangle=\sum_n c_n|a_n\rangle\) measured in the \(|a_n\rangle\) basis.

A state is \(|\psi\rangle=(3/5)|a_1\rangle+(4/5)|a_2\rangle\). Find the probabilities of the two outcomes.

A state is \(|\psi\rangle=C(|a_1\rangle+2i|a_2\rangle)\). Find \(C\) for normalization and the outcome probabilities.

For outcomes \(a_1=2\) and \(a_2=5\) with probabilities \(1/4\) and \(3/4\), find \(\langle A\rangle\).

Using the same outcomes \(2\) and \(5\) with probabilities \(1/4\) and \(3/4\), find \(\Delta A\).

What is the state immediately after a nondegenerate ideal measurement gives result \(a_k\)?

Why does immediately repeating the same ideal nondegenerate measurement give the same result?

A state is an eigenstate of \(\hat A\), but a measurement of \(\hat B\) is made instead. What determines the probabilities of \(B\) outcomes?

For a spin-half state \(|+x\rangle=(|+z\rangle+|-z\rangle)/\sqrt2\), find the probabilities for measuring \(S_z\).

A state is \((|E_1\rangle+|E_2\rangle)/\sqrt2\). What are the possible energy measurement results and their probabilities?

For \((|E_1\rangle+|E_2\rangle)/\sqrt2\), what state remains after measuring energy and obtaining \(E_2\)?

Explain why an expectation value need not be a possible single measurement result.

What does it mean for two observables to be compatible in the ideal operator model?

If \([\hat A,\hat B]\ne0\), why can measuring \(A\) disturb later measurements of \(B\)?

A measurement has a degenerate eigenvalue \(a\) with two orthonormal eigenstates \(|a,1\rangle\) and \(|a,2\rangle\). If the initial state is \(c_1|a,1\rangle+c_2|a,2\rangle+c_3|b\rangle\), what is the probability of result \(a\)?

For a continuous position measurement, how is the probability of finding the particle in an interval related to \(\psi(x)\)?

Prove that Born-rule probabilities sum to one when the state is normalized and expanded in an orthonormal measurement basis.

Distinguish the pure state \((|1\rangle+|2\rangle)/\sqrt2\) from a 50-50 classical mixture of \(|1\rangle\) and \(|2\rangle\).

A spin-half system starts in \(|+z\rangle\). Measure \(S_x\), ignore the result, then measure \(S_z\). Show that the final \(S_z\) probabilities are equal.