EigenstateEdit
An eigenstate is a fundamental concept in linear algebra and quantum theory. In its simplest mathematical form, an eigenstate is a vector v that, when acted on by a linear operator A, is scaled by a constant factor λ: Av = λv. In quantum mechanics, states of a system live in a Hilbert space, and observables are represented by linear operators on that space. An eigenstate of a given observable therefore yields a definite value for that measurement: if the system is in an eigenstate v associated with eigenvalue λ, a measurement of the corresponding observable will yield λ with certainty (up to the usual probabilistic rules that apply when the state is not exactly an eigenstate). The topic sits at the intersection of mathematics, physics, and applied science, and it underpins how we understand and predict the behavior of physical systems.
In quantum theory, the most important operators are Hermitian (or self-adjoint) operators, since these have real eigenvalues and orthogonal eigenvectors. The eigenvectors form a basis for the part of the Hilbert space relevant to the observable, enabling a diagonal representation of the operator. This diagonalization is what makes many problems tractable: in the eigenbasis, the action of the operator is simply to rescale each basis state by its eigenvalue. The probabilistic structure of quantum mechanics—the Born rule—tells us how amplitudes in a superposition relate to measurement outcomes, and the projection postulate describes how the state updates after a measurement that yields a particular eigenvalue.
Mathematical background
- An eigenstate is a special kind of eigenvector. For a square matrix or a linear operator A, an eigenvector v and its eigenvalue λ satisfy Av = λv. In a quantum context, A is often an observable such as the Hamiltonian, the momentum operator, or the angular momentum operator. See Linear operator and Hermitian operator.
- In finite dimensions, a Hermitian operator has a complete set of real eigenvalues and an orthonormal set of eigenvectors. This yields a convenient basis for expanding any state. See Eigenvalue and Eigenvector.
- In infinite dimensions, the situation generalizes via the spectral theorem. Observables may have discrete spectra (countable eigenvalues) and/or continuous spectra, and eigenstates can form a basis only for the discrete part of the spectrum. See Hilbert space and Spectral theorem.
- Time evolution in quantum mechanics is governed by the Schrödinger equation; eigenstates of the Hamiltonian are called stationary states because their time dependence is only a phase factor, leaving observables' probabilities unchanged in time. See Schrödinger equation and Hamiltonian.
In quantum mechanics
- An energy eigenstate is an eigenstate of the Hamiltonian operator Hamiltonian with eigenvalue E. The state evolves as |ψ(t)⟩ = e^{-iEt/ħ}|ψ(0)⟩, up to an overall phase, which does not affect measurement statistics. See Energy eigenstate.
- Measuring an observable corresponding to a Hermitian operator yields one of its eigenvalues with probability given by the squared magnitude of the projection of the state onto the corresponding eigenstate, per the Born rule. After the measurement, the system collapses into the associated eigenstate (or, for degeneracies, into the corresponding subspace). See Born rule and Projection postulate.
- A complete set of eigenstates allows the expansion of any state in that observable’s basis. This is central to solving many quantum problems by transforming to a basis where the operator acts diagonally. See Completeness relation.
Examples
- Spin-1/2 systems: the spin along a chosen axis (for example, the z-axis) has eigenstates |↑⟩ and |↓⟩ with eigenvalues +ħ/2 and −ħ/2 for the S_z operator, respectively. These eigenstates form the basis for describing any spin state via superpositions. See Spin (quantum number) and Angular momentum.
- Particle in a one-dimensional box: the Hamiltonian has a discrete set of energy eigenfunctions that satisfy boundary conditions, providing a clear illustration of how eigenstates encode fixed energy values for a given potential. See Particle in a box.
- Hydrogen atom: the stationary states are eigenstates of the hydrogenic Hamiltonian, characterized by quantum numbers that label energy and angular properties. These eigenstates underpin spectroscopy and chemical behavior. See Hydrogen atom.
- Crystalline solids: electrons in a periodic potential have Bloch states, which are eigenstates of the crystal momentum operator in the reduced Brillouin zone. These ideas connect to electronic structure and transport properties. See Bloch state and Crystalline solids.
Role in interpretation and debates
- Operational and pragmatic view: in practice, physicists rely on eigenstates as a calculational tool to predict outcomes of measurements and to design experiments or devices. The mathematics provides reliable, testable predictions even if one debates the deeper meaning of the wavefunction.
- Interpretative debates: among foundations of quantum mechanics, there are competing views about what the wavefunction and eigenstates represent. The Copenhagen interpretation emphasizes operational predictions and the role of measurement, while the Many-Worlds interpretation and other approaches raise questions about the nature of reality beyond what is observed. These debates are ongoing in philosophy of physics, but they rarely affect the standard use of eigenstates in technology, chemistry, and engineering.
- From a practical standpoint, controversies about interpretation should not obscure the fact that the formalism—eigenstates, eigenvalues, and their associated probabilities—has yielded reliable technologies, from semiconductors to MRI machines to quantum simulators. Critics who argue that such debates delegitimize established science often miss how operational success informs policy and industry, while defenders of the practical approach emphasize that scientific progress has historically advanced by solving real problems rather than by dwelling on metaphysical questions.