Quantum measurement is interesting.
To begin with, take a hermitian operator $H$. $|\psi\rangle$ is said to be an eigenvector of $H$ with eigenvalue $\lambda$ if $H|\psi\rangle=\lambda|\psi\rangle$. Hermitian operators have the nice property:
Theorem. All eigenvalues of a hermitian operator are real. Let $|\psi\rangle$ be an eigenvector of $H$ with eigenvalue $\lambda$. Then $\langle\psi|H|\psi\rangle$ is real, and at the same time $\langle\psi|H|\psi\rangle=\langle\psi|\lambda|\psi\rangle=\lambda\langle\psi|\psi\rangle$. Now $\langle\psi|\psi\rangle$ is real, hence $\lambda$ is real.
For an eigenvalue $\lambda$, we define the projection operator $\Pi_\lambda$ as the unique operator such that:
- $\Pi_\lambda|\psi\rangle=|\psi\rangle$ if $|\psi\rangle$ is an eigenvector of $H$ with eigenvalue $\lambda$.
- $\Pi_\lambda|\psi\rangle=0$ if $|\psi\rangle$ is an eigenvector of $H$ with eigenvalue different from $\lambda$.
Nicely, the projection operators are well defined for any hermitian operator. To wee why they aren’t always defined for non-hermitian operators, consider the operator $O|1\rangle=|1\rangle+|0\rangle$ and $O|0\rangle=|0\rangle$, and try to compute $\Pi_1|1\rangle$.
Now we can describe what happens when we perform a measurement of an operator $H$ on a state $|\psi\rangle$:
- The measurement returns one of the eigenvalues of $H$. The probability of returning $\lambda$ is the length of $\Pi_\lambda|\psi\rangle$.
- The state collapses to $\Pi_\lambda|\psi\rangle$, suitably scaled to have length 1.
The collapsing rule is probably the most confusing part of quantum mechanics, period. Think about it this way: if you make two measurements of the same quantity immediately after one another, then you expect the same value each time. However, without the collapsing rule, you might get different values each time. Not that this does any justice to the collapsing rule, in fact, a large part of the difficulty in interpreting quantum mechanics is the debate over how to think of the collapsing rule.
A concrete example is due. Take the operator $H|0\rangle=|0\rangle+2|1\rangle$ and $H|1\rangle=2|0\rangle+|1\rangle$. The eigenvalues and eigenvectors are $|0\rangle+|1\rangle$ with eigenvalue 3 and $|0\rangle-|1\rangle$ with eigenvalue -1. Now measure $H$ on the state $|0\rangle$. This has $\Pi_3|0\rangle=\frac{1}{2}(|0\rangle+|1\rangle)$ and $\Pi_{-1}|0\rangle=\frac{1}{2}(|0\rangle-|1\rangle)$. Their lengths are $\frac{1}{\sqrt 2}$, so the measurement has a $\frac{1}{2}$ chance of either value. If the value end up as 3, then the resulting state is a scaled version of $\frac{1}{2}(|0\rangle+|1\rangle)$, which is $\frac{1}{\sqrt 2}(|0\rangle+|1\rangle)$.