Introduction: What Operators Do
An operator \(\hat{A}\) is a function that maps states to states: \(\hat{A}|\psi\rangle = |\psi'\rangle\). In a finite-dimensional Hilbert space with a chosen basis, every operator can be represented as a matrix. In infinite dimensions, operators are more subtle but the matrix analogy still guides our intuition.
Matrix Elements
Given an orthonormal basis \(\{|n\rangle\}\), the matrix elements of operator \(\hat{A}\) are:
\[A_{mn} = \langle m|\hat{A}|n\rangle\]The matrix representation is:
\[A = \begin{pmatrix} \langle 1|\hat{A}|1\rangle & \langle 1|\hat{A}|2\rangle & \cdots \\ \langle 2|\hat{A}|1\rangle & \langle 2|\hat{A}|2\rangle & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}\]Operator Algebra
- Sum: \((\hat{A} + \hat{B})|\psi\rangle = \hat{A}|\psi\rangle + \hat{B}|\psi\rangle\)
- Product: \((\hat{A}\hat{B})|\psi\rangle = \hat{A}(\hat{B}|\psi\rangle)\)
- Scalar multiple: \((c\hat{A})|\psi\rangle = c(\hat{A}|\psi\rangle)\)
- Adjoint: \(\langle\phi|\hat{A}|\psi\rangle = \langle\hat{A}^\dagger\phi|\psi\rangle\)
Worked Examples
Example 1: Matrix of the Pauli X Operator
In the \(\{|0\rangle, |1\rangle\}\) basis, if \(\hat{\sigma}_x|0\rangle = |1\rangle\) and \(\hat{\sigma}_x|1\rangle = |0\rangle\):
\[\sigma_x = \begin{pmatrix} \langle 0|\hat{\sigma}_x|0\rangle & \langle 0|\hat{\sigma}_x|1\rangle \\ \langle 1|\hat{\sigma}_x|0\rangle & \langle 1|\hat{\sigma}_x|1\rangle \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]Example 2: Applying an Operator
Apply \(\hat{\sigma}_x\) to \(|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\):
\[\hat{\sigma}_x|\psi\rangle = \frac{1}{\sqrt{2}}(\hat{\sigma}_x|0\rangle + \hat{\sigma}_x|1\rangle) = \frac{1}{\sqrt{2}}(|1\rangle + |0\rangle) = |\psi\rangle\]The state is unchanged—it's an eigenstate with eigenvalue 1!
Example 3: Computing a Matrix Element
For the lowering operator \(\hat{a}\) with \(\hat{a}|n\rangle = \sqrt{n}|n-1\rangle\):
\[\langle m|\hat{a}|n\rangle = \sqrt{n}\langle m|n-1\rangle = \sqrt{n}\delta_{m,n-1}\]Non-zero only when \(m = n-1\).
The Quantum Connection
In quantum mechanics, physical quantities (position, momentum, energy) are represented by operators. The expectation value is:
\[\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle\]This is the average value you'd measure over many identical experiments. The matrix representation makes calculation concrete: it's just matrix-vector multiplication.