What Is a Stationary State?

A stationary state is a quantum mechanical state in which all observable physical properties of a system are constant in time. The term does not imply that the system is motionless; rather, the probability density associated with the state, which determines the likelihood of measuring any observable quantity at any position, does not evolve. Stationary states are the solutions to the time-independent Schrödinger equation, the central eigenvalue equation of non-relativistic quantum mechanics, and they correspond physically to states of definite energy.

The concept is foundational to quantum theory. When a quantum system is prepared in a stationary state, repeated measurements of the total energy always return the same value, and no observable quantity changes with time. This stands in sharp contrast to a general quantum superposition, in which different energy components oscillate at different frequencies and the probability density evolves in a complex, time-dependent pattern.

Energy Eigenstates and the Schrödinger Equation

A stationary state is formally defined as an eigenstate of the Hamiltonian operator, the quantum mechanical operator representing total energy. The time-independent Schrödinger equation, written as H ψ = E ψ, determines the allowed energy eigenvalues E and the corresponding eigenfunctions ψ for a system with a given potential. The full time-dependent wavefunction of a stationary state takes the form ψ(x,t) = ψ(x) exp(−iEt/ℏ), where ℏ is the reduced Planck constant. Because the exponential factor has unit magnitude, the probability density |ψ(x,t)|² reduces to |ψ(x)|², which is independent of time.

As explained in the Physics LibreTexts treatment of stationary states in introductory quantum mechanics, this time-independence of the probability density is the defining characteristic that earns these states the name "stationary." The spatial wavefunction ψ(x) encodes the quantum number structure of the state: for the hydrogen atom, it specifies the principal, angular, and magnetic quantum numbers.

Physical Properties and Constants of Motion

In a stationary state, any observable whose corresponding operator commutes with the Hamiltonian and carries no explicit time dependence will have a constant expectation value and zero variance over time. This makes stationary states the natural reference states for computing atomic and molecular properties, including ionization energies, transition dipole moments, and selection rules. The energy levels of the hydrogen atom, calculated from the eigenvalues of the Coulomb Hamiltonian, directly predict the spectral lines observed in emission and absorption spectroscopy, a connection confirmed experimentally long before quantum mechanics was formally developed.

For bound systems such as the particle in a box, the harmonic oscillator, or the hydrogen atom, the allowed energies form a discrete spectrum. Each level corresponds to one (or, with degeneracy, several) stationary state. As discussed in the University of Texas quantum mechanics lecture notes on stationary states, the ground state is the stationary state with the lowest allowed energy, and it is the state in which an isolated quantum system settles after energy dissipation.

Superposition and Time Evolution

A general quantum state is expressed as a linear combination of stationary states, each weighted by a complex coefficient. When such a superposition evolves in time, the different energy components accumulate phase at different rates, producing oscillations in the probability density at frequencies proportional to the energy differences between levels. The stationary states themselves, however, remain unchanged. The completeness of the energy eigenstate basis means that any physically realizable quantum state can be decomposed this way, making the MIT OpenCourseWare treatment of energy eigenstates and stationary states a standard reference for understanding quantum dynamics.

Applications

The stationary state concept has applications in a wide range of areas, including:

• Atomic and molecular spectroscopy, predicting emission and absorption line positions
• Quantum chemistry and the calculation of electronic structure in molecules
• Semiconductor physics, describing the quantized energy levels in quantum wells and dots
• Laser physics, defining the upper and lower levels between which stimulated emission occurs
• Quantum computing, where energy eigenstates serve as computational basis states