Introduction: Directions That Don't Rotate
When a matrix acts on most vectors, it changes both their magnitude and direction. But some special vectors only get stretched (or compressed)—their direction stays the same. These are eigenvectors, and their stretch factors are eigenvalues.
Definition
A nonzero vector \(|\psi\rangle\) is an eigenvector of operator \(\hat{A}\) with eigenvalue \(\lambda\) if:
\[\hat{A}|\psi\rangle = \lambda|\psi\rangle\]The operator just scales the vector—it doesn't change its direction.
Why Eigenvalues Matter
- In quantum mechanics, eigenvalues are the possible measurement outcomes
- Eigenvectors are the states of definite value
- Eigenvalue equations define stationary states of time evolution
- They simplify matrix operations: \(A^n|\psi\rangle = \lambda^n|\psi\rangle\)
Worked Examples
Example 1: Pauli Z Matrix
Find eigenvalues and eigenvectors of \(\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\):
For \(|+\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\): \(\sigma_z|+\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} = 1 \cdot |+\rangle\)
Eigenvalue: \(\lambda = +1\)
For \(|-\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\): \(\sigma_z|-\rangle = \begin{pmatrix} 0 \\ -1 \end{pmatrix} = -1 \cdot |-\rangle\)
Eigenvalue: \(\lambda = -1\)
Example 2: Pauli X Matrix
\(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
The eigenvectors are \(|+_x\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}\) (eigenvalue +1) and \(|-_x\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) (eigenvalue -1).
Example 3: Non-Eigenvector
Is \(|\psi\rangle = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\) an eigenvector of \(\sigma_z\)?
\[\sigma_z|\psi\rangle = \begin{pmatrix} 1 \\ -1 \end{pmatrix} \neq \lambda\begin{pmatrix} 1 \\ 1 \end{pmatrix}\]No—the direction changed.
The Quantum Connection
When you measure an observable \(\hat{A}\), the only possible results are its eigenvalues. If the system is in eigenstate \(|a\rangle\), you get eigenvalue \(a\) with certainty. If it's in a superposition, the eigenvalue equation determines which outcomes are possible and measurement collapses the state to the corresponding eigenvector.