Introduction: The Central Theorem of Quantum Measurement
The Spectral Theorem states that Hermitian operators can always be diagonalized by a unitary transformation, with real eigenvalues and orthonormal eigenvectors. This is the mathematical foundation for quantum measurement.
Statement of the Theorem
If \(\hat{A} = \hat{A}^\dagger\) (Hermitian), then:
- All eigenvalues are real
- Eigenvectors for distinct eigenvalues are orthogonal
- There exists an orthonormal basis of eigenvectors
- \(\hat{A}\) can be written as: \(\hat{A} = \sum_n a_n |a_n\rangle\langle a_n|\)
Proof of Real Eigenvalues
If \(\hat{A}|a\rangle = a|a\rangle\), take the inner product with \(\langle a|\):
\[\langle a|\hat{A}|a\rangle = a\langle a|a\rangle = a\]Now use Hermiticity: \(\langle a|\hat{A}|a\rangle = \langle \hat{A}a|a\rangle = \overline{\langle a|\hat{A}^\dagger|a\rangle} = \overline{\langle a|\hat{A}|a\rangle} = \bar{a}\)
Therefore \(a = \bar{a}\), so \(a\) is real. ∎
Worked Examples
Example 1: Spectral Decomposition of σ_z
\[\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = 1 \cdot |+\rangle\langle +| + (-1) \cdot |-\rangle\langle -|\] \[= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \checkmark\]Example 2: Spectral Decomposition of σ_x
Eigenvalues: ±1. Eigenvectors: \(|\pm_x\rangle = \frac{1}{\sqrt{2}}(|+\rangle \pm |-\rangle)\)
\[\sigma_x = (+1)|+_x\rangle\langle +_x| + (-1)|-_x\rangle\langle -_x|\]Example 3: Orthogonality Check
For \(\sigma_z\): \(\langle +|-\rangle = (1, 0)\begin{pmatrix} 0 \\ 1 \end{pmatrix} = 0\) ✓
Eigenvectors with eigenvalues +1 and -1 are orthogonal.
The Quantum Connection
The Spectral Theorem is why quantum mechanics works:
- Real eigenvalues → Measurement results are real numbers
- Orthogonal eigenvectors → Distinct outcomes are mutually exclusive
- Complete basis → Any state can be expanded, so probabilities sum to 1
Every observable satisfies these requirements because every observable is Hermitian.