Eigenvalue Equations

Introduction

Applying an operator to a wavefunction usually produces a different function. Special wavefunctions return their original shape multiplied by a number. These are eigenfunctions, and the numbers are eigenvalues. In quantum chemistry the distinction matters because an observable such as energy has definite measured values in its eigenstates. An arbitrary wavefunction can be a combination of such states and need not have one definite outcome.

Core explanation

For an operator Â, an eigenvalue equation is Âψ = aψ, where ψ is a nonzero function and a is a constant independent of position. The operator acts on ψ; if the result is the same function times a, ψ is an eigenfunction and a its eigenvalue. For a physical observable, the operator has the appropriate Hermitian property and allowed eigenvalues are real under correct boundary and domain conditions. Merely seeing a factor that depends on x does not satisfy the equation because an eigenvalue must be constant for that state.

Take the momentum operator p̂ₓ = −iħ d/dx. Acting on e^(ikx) gives −iħ(ik)e^(ikx) = ħk e^(ikx). The plane wave is a formal momentum eigenfunction with eigenvalue ħk, though a plane wave across all space is not normalizable as a single-particle probability distribution and is treated with suitable idealization. Acting on sin(kx) instead gives −iħk cos(kx), not a constant times sin(kx), so that sine is not a momentum eigenfunction on its own.

An operator may have several eigenfunctions with the same eigenvalue; this is degeneracy. Also, a state can be written as a superposition of eigenfunctions, ψ = c₁φ₁ + c₂φ₂, with Âφᵢ = aᵢφᵢ. If a₁ ≠ a₂, applying  yields c₁a₁φ₁ + c₂a₂φ₂, not generally a single constant times ψ. A measurement of A can produce the allowed eigenvalues, with probabilities determined by appropriately normalized expansion coefficients under the measurement postulates. The expectation value is an average over many identically prepared measurements, not necessarily one allowed individual outcome.

Boundary conditions matter. The same differential operator can have different allowed eigenfunctions and discrete or continuous spectra depending on the physical region and conditions. A particle in an infinite box has quantized standing-wave energy eigenfunctions because wavefunctions vanish at the walls. A free particle is described by a continuum of momentum-like states. Thus an operator symbol alone does not determine all eigenvalues; the Hamiltonian, domain and boundaries complete the problem.

MIT OpenCourseWare physical chemistry lecture notes at https://ocw.mit.edu/courses/5-61-physical-chemistry-fall-2007/pages/lecture-notes/ develop operators, eigenstates and model systems. The practical chemistry link is spectroscopy: only differences between allowed energy eigenvalues can appear as transition energies under the relevant selection rules.

Step-by-step reasoning

1. Identify the operator and its domain or boundary conditions. 2. Apply it to the proposed nonzero function without skipping derivatives. 3. Divide the result by the original function only where mathematically valid. 4. Check whether the ratio is one constant rather than a coordinate-dependent expression. 5. Interpret that constant as a possible observable value when the operator represents a physical quantity.

Visual explanation

Draw a machine labeled  receiving ψ. One output has exactly the same curve shape but scaled vertically by a, so it is an eigenfunction. A second output has changed shape, so no single eigenvalue describes that input. Add an energy-level ladder beside a Hamiltonian machine to show that its eigenvalues correspond to allowed energies.

Real-world analogy

A musical instrument can vibrate in special normal modes. Exciting one mode reproduces its shape while its amplitude changes; exciting several modes creates a more complicated pattern. Operator eigenfunctions play a related mathematical role, though quantum measurement probabilities are not simply classical sound amplitudes.

Real-world example

Hydrogen's orbitals are eigenfunctions of its idealized nonrelativistic Hamiltonian. Their energies are eigenvalues. A laboratory sees photons at energies corresponding to differences between these levels, not a continuous spread of every imaginable electron energy for isolated bound-state transitions.

Why?

Why is the eigenfunction condition useful for measurement? A state that is an eigenstate of an observable has a definite outcome for that observable in the ideal model. If the state is a superposition of different eigenvalues, repeated measurements can yield different allowed outcomes according to the state's components.

Common misconception

“Any function gives an eigenvalue when an operator acts on it” is false. The result must be a constant multiple of the original function. Another mistake is identifying the expectation value with an outcome that must occur in each single measurement; an average can lie between discrete allowed eigenvalues.

Worked example

Let D = d/dx and f(x) = e^(3x). Then Df = 3e^(3x) = 3f, so f is an eigenfunction of D with eigenvalue 3 on a suitable mathematical domain. Let g(x) = x². Then Dg = 2x, which is not a constant times x² for all x, so g is not an eigenfunction of D. The same test applies to physical operators, with additional boundary and normalizability requirements.

Quick check

1. If Âψ = xψ, is ψ an eigenfunction with eigenvalue x? Answer: Not generally. An eigenvalue for a given state must be a constant; x varies with position. Only a special domain or distributional interpretation could change that conclusion.

Exam focus

Apply the operator first, then test proportionality. State the constant eigenvalue and units appropriate to the observable. Distinguish a formal plane-wave eigenfunction from a normalized bound-state wavefunction. When discussing measurement, distinguish definite eigenstate outcome from superposition probabilities and expectation values.

Advanced insight

Commuting operators can possess shared eigenfunctions under suitable conditions, allowing several quantum numbers to label a state. Noncommuting operators generally do not admit a complete common eigenbasis. In hydrogen, the Hamiltonian and angular-momentum operators can be used together to label idealized stationary states, making n, l and mₗ more than arbitrary names.

Summary

An eigenvalue equation returns the original function multiplied by a constant. Eigenvalues of observable operators are possible measurement outcomes, subject to domains and boundary conditions. Superpositions of different eigenstates generally lack a single definite value, while expectation values describe ensemble averages.

Practice questions

1. Is cos(kx) an eigenfunction of d²/dx²? Give the eigenvalue. Answer: Yes. Its second derivative is −k²cos(kx), so the eigenvalue of d²/dx² is −k².

2. Is sin(kx) an eigenfunction of −iħ d/dx by itself? Answer: No. The derivative is proportional to cos(kx), not to the original sine function.

3. If two energy eigenstates have different energies, can their nontrivial sum have one definite energy? Answer: Generally no. Applying the Hamiltonian multiplies the components by different constants, so the sum is not one energy eigenfunction.