Introduction
Eigenvalues and eigenfunctions form the mathematical bedrock of quantum mechanics. Operators represent observables. Eigenvalues correspond to measurable quantities. Eigenfunctions describe quantum states with definite measurement outcomes. This framework enables prediction of experimental results and underpins quantum theory.
"The quantum state is described by a wavefunction which is an eigenfunction of an observable operator, revealing the quantized nature of physical quantities." -- P. A. M. Dirac
Operators in Quantum Mechanics
Definition of Operators
Operators: linear mappings on Hilbert space. Act on state vectors (wavefunctions). Represent physical observables such as position, momentum, energy.
Types of Operators
Hermitian (self-adjoint) operators: correspond to measurable quantities. Unitary operators: preserve norm, represent transformations. Projection operators: represent measurement outcomes.
Operator Properties
Linearity: \( \hat{O}(a\psi + b\phi) = a\hat{O}\psi + b\hat{O}\phi \). Adjoint: \( \hat{O}^\dagger \) defined by \( \langle \phi | \hat{O}\psi \rangle = \langle \hat{O}^\dagger \phi | \psi \rangle \). Hermitian: \( \hat{O} = \hat{O}^\dagger \).
Eigenvalue and Eigenfunction: Definition
Mathematical Formulation
Given operator \( \hat{O} \), eigenvalue \( \lambda \), eigenfunction \( \psi \):
\hat{O} \psi = \lambda \psiEigenfunction: nonzero function satisfying above. Eigenvalue: scalar associated to eigenfunction.
Hilbert Space Context
Eigenfunctions belong to Hilbert space \( \mathcal{H} \). Usually normalized: \( \langle \psi | \psi \rangle = 1 \). Eigenvalues are real for Hermitian operators.
Spectrum Classification
Discrete spectrum: isolated eigenvalues. Continuous spectrum: ranges of values. Residual spectrum: pathological cases excluded from physical operators.
Hermitian Operators and Spectrum
Hermiticity and Reality of Eigenvalues
Hermitian operators guarantee real eigenvalues. Proof via inner product symmetry:
\( \langle \psi | \hat{O} \psi \rangle = \langle \hat{O} \psi | \psi \rangle \Rightarrow \lambda = \lambda^* \)
Orthogonality of Eigenfunctions
Eigenfunctions of distinct eigenvalues are orthogonal:
\( \langle \psi_m | \psi_n \rangle = 0 \) if \( \lambda_m \neq \lambda_n \).
Completeness
Set of eigenfunctions forms a complete basis in \( \mathcal{H} \). Any state expressible as linear combination of eigenfunctions.
Physical Interpretation
Observables and Measurement
Operators represent physical observables. Eigenvalues represent possible measurement results. Eigenfunctions represent states with definite outcomes.
Collapse Postulate
Measurement collapses wavefunction to eigenfunction associated with observed eigenvalue. Probability given by projection squared.
Expectation Values
Expectation value for observable \( \hat{O} \) in state \( \psi \):
\( \langle \hat{O} \rangle = \langle \psi | \hat{O} | \psi \rangle \).
Measurement Postulate
Postulate Statement
Measurement outcome is eigenvalue \( \lambda \) of observable operator \( \hat{O} \). Result probabilistic unless state is eigenfunction.
Probability Rule
Probability of measuring \( \lambda \) when system in state \( \phi \):
\( P(\lambda) = |\langle \psi_\lambda | \phi \rangle|^2 \) where \( \psi_\lambda \) eigenfunction.
Post-Measurement State
State collapses to eigenfunction \( \psi_\lambda \) corresponding to measurement outcome.
Schrödinger Equation Application
Time-Independent Schrödinger Equation
Eigenvalue problem for Hamiltonian operator \( \hat{H} \):
\( \hat{H} \psi = E \psi \), where \( E \) is energy eigenvalue.
Stationary States
Eigenfunctions \( \psi \) are stationary states with definite energies. Time evolution:
\( \Psi(x,t) = \psi(x) e^{-iEt/\hbar} \).
Energy Quantization
Discrete eigenvalues correspond to quantized energy levels. Basis for atomic spectra and quantum stability.
Degeneracy and Orthogonality
Degeneracy
Multiple eigenfunctions share same eigenvalue. Degeneracy arises from symmetry or conserved quantities.
Orthogonality Within Degenerate Subspace
Degenerate eigenfunctions can be chosen orthogonal. Gram-Schmidt procedure applicable.
Physical Implications
Degeneracy linked to conserved quantum numbers. Splitting via perturbations breaks degeneracy (lifting).
Spectral Decomposition Theorem
Theorem Statement
Hermitian operator \( \hat{O} \) can be decomposed as:
\( \hat{O} = \sum_n \lambda_n |\psi_n \rangle \langle \psi_n| \) discrete,
or integral over continuous spectrum.
Projection Operators
Each \( |\psi_n \rangle \langle \psi_n| \) is projection operator onto eigenspace.
Application to Quantum Dynamics
Used in propagators, time evolution, and measurement theory.
| Operator Type | Spectral Decomposition |
|---|---|
| Discrete Spectrum | Sum over eigenvalues and projectors |
| Continuous Spectrum | Integral over spectral measure |
Examples of Operators
Position Operator \( \hat{x} \)
Acts multiplicatively: \( \hat{x} \psi(x) = x \psi(x) \). Eigenfunctions: delta functions \( \delta(x - x_0) \). Eigenvalues: position \( x_0 \).
Momentum Operator \( \hat{p} \)
Defined by \( \hat{p} = -i \hbar \frac{d}{dx} \). Eigenfunctions: plane waves \( e^{ipx/\hbar} \). Eigenvalues: momentum \( p \).
Hamiltonian Operator \( \hat{H} \)
Energy operator, typically \( \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) \). Eigenvalues: energy levels. Eigenfunctions: stationary states.
Numerical Methods
Matrix Representation
Operators represented as matrices in finite basis. Eigenvalue problems reduce to matrix diagonalization.
Common Algorithms
Power iteration, QR algorithm, Lanczos method. Efficient for sparse or large matrices.
Applications
Computing energy spectra, simulating quantum systems, solving Schrödinger equation numerically.
Algorithm: Power IterationInput: matrix A, initial vector v0Repeat: v_{k+1} = A v_k / ||A v_k||Until convergenceOutput: dominant eigenvalue and eigenvectorSummary
Eigenvalues and eigenfunctions connect quantum observables and measurable quantities. Hermitian operators ensure real eigenvalues and orthogonal eigenfunctions. Measurement collapses states onto eigenfunctions with probabilities linked to projections. Spectral theorem underpins operator decompositions. Applications span energy quantization, dynamics, and numerical simulations.
References
- J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2010, pp. 45-89.
- R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, pp. 189-237.
- L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998, pp. 101-150.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, 1980, pp. 245-290.
- E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998, pp. 75-120.