Quantum measurements: Eigenvalue equations and projection equations

Question

In general, the action of an observable operator on a quantum state is a projection into one of the operator's eigenstates. However, the action of an operator on a state is written $$ \hat A|\psi\rangle= \hat A\sum c_n|a_n\rangle=\sum c_na_n|a_n\rangle .$$

This has not executed the projection operation. Namely, a measurement of $\hat A$ should be $$ \hat A|\psi\rangle\to|a_k\rangle ,$$

so that if eigenvalue $a_k$ is obtained from the first measurement, any number of rapidly repeated measurements will also yield $a_k$. The operator $\hat A$ has projected $\psi$ into the 1D eigenspace spanned by $|a_k\rangle$. Unfortunately, I have not written an equation that shows that. I have only used the $\to$ symbol to say "and this where the magic happens." Is it required that after finding eigenvalue $a_k$, I must operate with $\hat P_k(\psi)=\frac{1}{c_k}|a_k\rangle\langle a_k|$: $$ \hat P_k(\psi)|\psi\rangle=|a_k\rangle ?$$

That extra step seems cumbersome and clunky but I am not sure which is the notation I am looking for.

Q: What formalism is standard for the mathematical statement that the act of measurement collapses a state to an eigenstate: $$ f\big(|\psi\rangle\big)=|a_k\rangle? $$

Answer 1

The most general formalism to describe the action of a quantum measurement is the POVM formalism (which stands for 'positive operator-valued measure'). In its most general form, a measurement is described by a measurement operator $M_k$ via

$$|\psi_k\rangle \rightarrow \frac{M_k |\psi\rangle}{\sqrt{\langle\psi|M^\dagger M|\psi\rangle}}$$

$M_k^\dagger M_k$ is called the POVM, which must satisfy the normalization condition

$$\sum_k M_k^\dagger M_k = I$$

The only difference between this and what you wrote is that the measurement operator does not contain the state-dependent normalization factor. For standard projective measurement of a Hermitian operator like your $\hat{A}$, we just take $M_k = |a_k\rangle\langle a_k|$, where $|a_k\rangle$ are the normalized eigenvectors of $\hat{A}$.

All of the above isn't really different from what you wrote at the end of the day, and there's no need to distinguish $M_k$ from $M_k^\dagger M_k$ for projective measurements. But it let's you handle measurements that can't be described as perfect projections, such as weak measurements. It's also nice that it describes time evolution by taking $M_0 = U$ where $U$ is unitary. So it unifies the two main postulates of quantum mechanics into one equation.