Introduction: Collapsing Onto Subspaces
A projection operator extracts the component of a vector along a specified direction or subspace. In quantum mechanics, projectors represent measurement—they collapse superpositions onto definite outcomes.
Properties of Projectors
An operator \(\hat{P}\) is a projector if:
- Idempotent: \(\hat{P}^2 = \hat{P}\) (projecting twice = projecting once)
- Hermitian: \(\hat{P}^\dagger = \hat{P}\) (for orthogonal projectors)
For a normalized state \(|n\rangle\):
\[\hat{P}_n = |n\rangle\langle n|\]Resolution of the Identity
For a complete orthonormal basis \(\{|n\rangle\}\):
\[\sum_n |n\rangle\langle n| = \hat{I}\]This is the completeness relation—it says the projectors onto all basis states sum to the identity. We can insert this anywhere:
\[|\psi\rangle = \hat{I}|\psi\rangle = \sum_n |n\rangle\langle n|\psi\rangle = \sum_n c_n |n\rangle\]where \(c_n = \langle n|\psi\rangle\).
Worked Examples
Example 1: Projection onto a State
Project \(|\psi\rangle = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\) onto \(|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\):
\[\hat{P}_0|\psi\rangle = |0\rangle\langle 0|\psi\rangle = |0\rangle(1 \cdot 3 + 0 \cdot 4) = 3|0\rangle = \begin{pmatrix} 3 \\ 0 \end{pmatrix}\]Example 2: Verifying Idempotence
\[\hat{P}_0^2 = |0\rangle\langle 0|0\rangle\langle 0| = |0\rangle \cdot 1 \cdot \langle 0| = |0\rangle\langle 0| = \hat{P}_0 \checkmark\]Example 3: Completeness Check
In \(\mathbb{C}^2\):
\[|0\rangle\langle 0| + |1\rangle\langle 1| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \checkmark\]The Quantum Connection
Measurement in quantum mechanics is modeled by projection. If you measure observable \(\hat{A}\) and get eigenvalue \(a_n\), the state collapses:
\[|\psi\rangle \to \frac{\hat{P}_n|\psi\rangle}{\|\hat{P}_n|\psi\rangle\|} = \frac{|n\rangle\langle n|\psi\rangle}{|\langle n|\psi\rangle|}\]The probability of outcome \(n\) is \(|\langle n|\psi\rangle|^2 = \langle\psi|\hat{P}_n|\psi\rangle\).