Eigenvalues and eigenfunctions answer a central question in quantum mechanics: when you measure a physical observable, what values can you actually get, and what does the system look like when it has a definite value of that observable? These two concepts connect the abstract operator formalism of quantum mechanics to real, measurable quantities. They also provide the mathematical machinery for solving the Schrödinger equation and describing how quantum states evolve in time.

Eigenvalues and Eigenfunctions in Quantum Mechanics

Definition and Mathematical Representation

An eigenvalue equation has the general form:

$\hat{A}\psi = a\psi$

where $\hat{A}$ is an operator, $\psi$ is the eigenfunction, and $a$ is the eigenvalue. The operator acts on the function and returns the same function multiplied by a constant. That constant is the eigenvalue.

For the specific case of the Hamiltonian (the energy operator), this becomes the time-independent Schrödinger equation:

$\hat{H}\psi = E\psi$

Here $\hat{H}$ is the Hamiltonian operator, $\psi$ is the energy eigenfunction, and $E$ is the energy eigenvalue. Solving this equation for a given system means finding which functions $\psi$ satisfy it and which values of $E$ are allowed.

The concept isn't limited to energy. Every observable has its own operator, and each operator has its own set of eigenvalues and eigenfunctions. For example, the angular momentum operator $\hat{L}^2$ has eigenvalues $\ell(\ell+1)\hbar^2$ with corresponding eigenfunctions (the spherical harmonics).

The complete set of eigenfunctions for a given operator forms a basis for the Hilbert space of the system, meaning any valid quantum state can be written as a combination of those eigenfunctions.

Properties of Eigenvalues and Eigenfunctions

Orthonormality. Eigenfunctions of a Hermitian operator satisfy two conditions:

Orthogonal: The inner product of two different eigenfunctions is zero: $\langle\psi_i|\psi_j\rangle = 0$ for $i \neq j$
Normalized: The inner product of an eigenfunction with itself equals one: $\langle\psi_i|\psi_i\rangle = 1$

Together, these are written compactly as $\langle\psi_i|\psi_j\rangle = \delta_{ij}$ (the Kronecker delta).

Real eigenvalues. Hermitian operators (operators that equal their own adjoint, $\hat{A} = \hat{A}^\dagger$ ) always produce real eigenvalues. This is physically necessary because measurement outcomes must be real numbers. All quantum mechanical observables are represented by Hermitian operators for exactly this reason.

Degeneracy. Sometimes multiple linearly independent eigenfunctions share the same eigenvalue. This is called degeneracy. When it occurs, any linear combination of the degenerate eigenfunctions is also a valid eigenfunction for that eigenvalue. The hydrogen atom provides a familiar example: for $n = 2$ , there are four degenerate states (one 2s and three 2p orbitals) all sharing the same energy.

Significance of Eigenvalues and Eigenfunctions

Definition and Mathematical Representation, Amplification of quadratic Hamiltonians – Quantum

Solving the Schrödinger Equation

Finding the eigenvalues and eigenfunctions of the Hamiltonian is equivalent to solving the time-independent Schrödinger equation. Here's what each piece tells you:

Eigenvalues give the allowed energy levels of the system. Only these discrete (or continuous, depending on the system) values of energy can be measured.
Eigenfunctions describe the spatial distribution of the particle's wavefunction at each energy level. The shape of $\psi_n(x)$ tells you where the particle is likely to be found.

The general solution to the time-dependent Schrödinger equation is then built as a linear combination of these eigenfunctions:

$\Psi(x,t) = \sum_n c_n \psi_n(x) \, e^{-iE_n t/\hbar}$

The coefficients $c_n$ are determined by the initial conditions of the system, and from them you can calculate probability distributions and expectation values for any observable.

Quantum State Expansion and Time Evolution

Any quantum state $|\Psi\rangle$ can be expanded in the eigenbasis of an operator:

$|\Psi\rangle = \sum_n c_n |\psi_n\rangle$

The coefficients $c_n = \langle\psi_n|\Psi\rangle$ are probability amplitudes. The probability of measuring eigenvalue $a_n$ is $|c_n|^2$ , and these probabilities must sum to one: $\sum_n |c_n|^2 = 1$ .

For time evolution, the eigenfunctions themselves don't change, but each coefficient picks up a time-dependent phase factor $e^{-iE_n t/\hbar}$ . This is why energy eigenstates are called stationary states: a system in a single eigenstate only acquires an overall phase, so all measurable probabilities remain constant in time. Superpositions of different energy eigenstates, however, do evolve because the different phase factors cause interference patterns that shift over time.

Determining Eigenvalues and Eigenfunctions

Simple Quantum Mechanical Systems

Particle in a 1D infinite potential well ("particle in a box"):

Eigenvalues: $E_n = \frac{n^2 h^2}{8mL^2}$ , where $n = 1, 2, 3, \ldots$ , $m$ is the particle mass, and $L$ is the box width
Eigenfunctions: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$

These are sinusoidal standing waves with $n$ half-wavelengths fitting inside the box. Notice the energy scales as $n^2$ , so the spacing between levels increases with $n$ .

Quantum harmonic oscillator:

Eigenvalues: $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$ , where $n = 0, 1, 2, \ldots$ and $\omega$ is the classical angular frequency
Eigenfunctions: Products of Hermite polynomials and a Gaussian, $\psi_n(x) = N_n H_n(\alpha x) \, e^{-\alpha^2 x^2 / 2}$

The energy levels are evenly spaced by $\hbar\omega$ , and the ground state ( $n = 0$ ) has a nonzero zero-point energy of $\frac{1}{2}\hbar\omega$ .

Hydrogen atom:

Eigenvalues: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ , where $n = 1, 2, 3, \ldots$ is the principal quantum number
Eigenfunctions: Products of radial functions (associated Laguerre polynomials) and angular functions (spherical harmonics), characterized by quantum numbers $n$ , $\ell$ , and $m_\ell$

These eigenfunctions are the familiar atomic orbitals (1s, 2p, 3d, etc.).

Definition and Mathematical Representation, Liouville's theorem (Hamiltonian) - Wikipedia

Numerical Methods for Complex Systems

Most real systems can't be solved analytically. Three common numerical approaches:

Shooting method: You guess an eigenvalue, integrate the differential equation from one boundary, and check whether the solution satisfies the other boundary condition. Then you iteratively adjust the eigenvalue until it does.
Matrix diagonalization: Represent the Hamiltonian as a matrix in some chosen basis set. Diagonalizing this matrix yields the eigenvalues along the diagonal and the eigenvectors (which encode the eigenfunctions as linear combinations of basis functions).
Variational method: Construct a trial wavefunction with adjustable parameters and minimize the expectation value of energy, $\langle\hat{H}\rangle$ . The variational theorem guarantees this gives an upper bound to the true ground-state energy, so the lower you can push it, the closer you are to the exact answer.

Physical Meaning of Eigenvalues and Eigenfunctions

Measurement and Probability

The eigenvalues of an operator are the only possible outcomes you can get when measuring the corresponding observable. You never measure a value that isn't an eigenvalue.

If the system is in state $|\Psi\rangle = \sum_n c_n |\psi_n\rangle$ , the probability of obtaining eigenvalue $a_n$ upon measurement is:

$P(a_n) = |c_n|^2 = |\langle\psi_n|\Psi\rangle|^2$

The expectation value of an observable $\hat{O}$ in state $|\Psi\rangle$ is:

$\langle\hat{O}\rangle = \langle\Psi|\hat{O}|\Psi\rangle = \sum_n |c_n|^2 a_n$

This is the average value you'd obtain over many identical measurements on identically prepared systems.

The eigenfunctions themselves describe the spatial probability density through $|\psi(x)|^2$ , which gives the probability per unit length of finding the particle at position $x$ .

Quantum State Superposition and Interference

A quantum system doesn't have to be in a single eigenstate. It can exist in a superposition, where multiple eigenstates contribute simultaneously. The coefficients $c_n$ carry both magnitude and phase information, and this phase matters.

Constructive interference occurs where probability amplitudes add in phase, increasing $|\Psi|^2$
Destructive interference occurs where they cancel, reducing $|\Psi|^2$

Upon measurement, the superposition collapses to a single eigenstate corresponding to the measured eigenvalue. Immediately after measurement, the system is in that eigenstate with certainty.

Connection to Classical Mechanics

Quantum mechanics must reproduce classical results in the appropriate limit. Several ideas connect the two:

Bohr correspondence principle: As quantum numbers become very large, quantum mechanical predictions approach classical behavior. For the particle in a box, at large $n$ the probability density $|\psi_n(x)|^2$ oscillates so rapidly that its spatial average becomes uniform, matching the classical prediction.
Energy level spacing: The gap between adjacent eigenvalues shrinks relative to the total energy as quantum numbers increase, making the discrete spectrum effectively continuous.
Ehrenfest's theorem: The time derivatives of quantum expectation values $\langle x \rangle$ and $\langle p \rangle$ obey equations that mirror Newton's second law: $\frac{d\langle p \rangle}{dt} = -\left\langle \frac{\partial V}{\partial x} \right\rangle$ . This provides a direct bridge between quantum expectation values and classical equations of motion.

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