Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

Part I, Chapter 3 | Page 1 of 8

Eigenvalues & Spectral Theory

The spectrum of observables and measurement outcomes

3.1 Eigenvalue Problem

The eigenvalue equation is central to quantum mechanics:

$$\hat{A}|a\rangle = a|a\rangle$$

where $a$ is the eigenvalue and $|a\rangle$ is the corresponding eigenvector (or eigenstate).

Physical Interpretation:

When we measure an observable $\hat{A}$, we get one of its eigenvalues $a$. If the system is in eigenstate $|a\rangle$, the measurement always yields $a$ with certainty!

Discrete Spectrum

For bound systems (e.g., electron in atom), eigenvalues are discrete:

$$\hat{H}|n\rangle = E_n|n\rangle, \quad n = 0, 1, 2, \ldots$$

Continuous Spectrum

For unbound systems (e.g., free particle), eigenvalues form a continuum:

$$\hat{p}|p\rangle = p|p\rangle, \quad p \in \mathbb{R}$$

3.2 Spectral Theorem

For a Hermitian operator $\hat{A}$, eigenvectors corresponding to different eigenvalues are orthogonal:

$$\langle a' | a \rangle = 0 \quad \text{if } a' \neq a$$

Completeness Relation

For discrete spectrum, eigenstates form a complete orthonormal basis:

$$\sum_n |n\rangle\langle n| = \hat{I}$$

Any state can be expanded:

$$|\psi\rangle = \sum_n c_n |n\rangle = \sum_n |n\rangle\langle n|\psi\rangle$$

where $c_n = \langle n|\psi\rangle$ are probability amplitudes and $|c_n|^2$ is the probability of finding the system in state $|n\rangle$.

3.3 Example: Harmonic Oscillator

The quantum harmonic oscillator has Hamiltonian:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2$$

Eigenvalues (energy levels) are:

$$E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots$$

Note the zero-point energy $E_0 = \frac{1}{2}\hbar\omega$ even for the ground state! This is a purely quantum effect with no classical analog.

Key Concepts

  • • Eigenvalues of Hermitian operators are measurable values
  • • Eigenstates of different eigenvalues are orthogonal
  • • Completeness: any state expands in eigenbasis
  • • $|c_n|^2 = |\langle n|\psi\rangle|^2$ gives measurement probabilities
  • • Discrete spectrum: bound states; Continuous: unbound states
← Ch. 2: Linear Operators
Page 1 of 8
Chapter 3: Eigenvalues & Spectral Theory
Next: Ch. 4 →