Stationary States and Energy Eigenstates

The best clocks ever built are two stationary states in a box. An optical lattice clock holds $\sim10^4$ strontium atoms in a standing light wave and compares the phase clocks of two electronic eigenstates whose gap beats at $4.29\times10^{14}$ Hz; because both states are dynamical fixed points, the beat is reproducible to nineteen digits — a fractional stability such that two such clocks disagree by a second over the age of the universe. Every improvement is an exercise in this article's theorems: make the two states more stationary (suppress couplings that dress them with other energies) and the Bohr frequency becomes a better ruler. The breakdown of stationarity is equally important in modern physics. No excited atomic state is truly stationary — coupling to the electromagnetic vacuum gives it a width $\Gamma$ and a lifetime $\tau = \hbar/\Gamma$ — and the resulting Lorentzian linewidths are how astronomers weigh stellar atmospheres and how laser physicists engineer gain. The honest theory of those quasi-stationary states, complex energies and all, is downstream in Survival Amplitude and Decay of Unstable States.

The Time-Evolution Operator taught you to see dynamics as a rigid rotation of the state vector, generated by $\hat H$. Picture the flow on the unit sphere of Hilbert space: almost every state is swept along, tracing some complicated arc. But a rotation has axes. Certain rays are invariant under the flow — the state turns within its own ray, multiplying by a phase, going nowhere physically. Those invariant rays are the eigenkets of the generator. Dynamics does not merely happen to leave them unchanged; they are the invariant directions that organize the entire time evolution.

What does it feel like to be on a fixed ray? The state has a clock — its phase turns at the enormous rate $E/\hbar$ — but the clock is attached to nothing observable. Every prediction quantum mechanics can make compares the state against something, and the comparison always pairs the phase with its own conjugate. The phase evolves continuously, but without interference with another energy component it has no observable consequence.

Hold on to the last line of the figure. The phase of a single stationary state is unobservable — but phase differences between two of them are the most measured quantities in physics. Every spectral line is two stationary clocks compared. That comparison machinery — cross terms, beats, revivals — is deliberately not this article: the two-state beat was derived in the wave-mechanics twin, and the general engine is built next door in Spectral Decomposition of Time Evolution. This article owns the fixed points themselves.

The eigenvalue problem of the Hamiltonian — the time-independent Schrödinger equation, whose solving craft belongs to Time-Independent Schrödinger Equation and the whole one-dimensional chapter — selects the special kets:

where: $|n\rangle$ — an energy eigenket; $E_n$ — its eigenvalue, one entry of the energy menu. No time appears in (1); it is a structural statement about $\hat H$. Dynamics enters when The Time-Evolution Operator's spectral form acts on an eigenket: $\hat U(t)$ is built from $\hat H$, so its eigenvectors are the same kets, with exponentiated eigenvalues,

where: $e^{-iE_nt/\hbar}$ — a pure phase, modulus one. The state at time $t$ differs from the initial state only by a global phase. Since physical states are rays in Hilbert space, both vectors represent the same physical state: same ray, same state. Equation (2) says the eigenket is a fixed point of the physical dynamics. Energy is the rotation rate of an unreadable dial: an eigenstate at $1\ \mathrm{eV}$ turns its phase once every $4.14\ \mathrm{fs}$ — the clock computed in the QM0 Time-Dependent Schrödinger Equation — and equation (2) is why you never see it directly. The claim worth a derivation is the converse: phase-only evolution is not merely a perk of eigenstates, it is their fingerprint. Nothing else in Hilbert space evolves that gently.

The wave-mechanics twin proved the freeze the honest way: Stationary States and Separation of Variables cancelled the clocks inside $|\Psi|^2$, pushed them through $\hat Q$ for expectation values, and audited the energy spread to zero. Those derivations stand; we do not repeat them. What the ket register adds is reach: one line that freezes not just means and densities but the entire outcome distribution of every observable at once.

One classic subtlety survives every freeze theorem and deserves its own sentence: stationary does not mean motionless. A stationary state's statistics are rigid, but statistics can describe steady flow. A free particle's eigenstate $e^{ikx}$ carries the time-independent probability current $j = \hbar k/m$ — quote the current object from Probability Current in One Dimension, apply it to the plane wave, and a $25\ \mathrm{eV}$ electron beam flows at a permanent $2.97\times10^{6}\ \mathrm{m/s}$ while every one of its histograms stands still. A stationary state has time-independent probabilities, not necessarily vanishing probability current. (Bound states in one dimension escape this loophole for a structural reason: they can be chosen real, and real wavefunctions carry zero current — the reality corollary of Non-Degeneracy of One-Dimensional Bound States.)

When two eigenkets share an energy, $E_1 = E_2 = E$, the fixed-point structure thickens from rays to whole subspaces — with consequences for what "the" stationary states even are.

1 · The clock that turns $10^{33}$ times (confidence builder)

The hydrogen ground state has $E = -13.6\ \mathrm{eV}$. Find the phase angular rate, the number of full phase turns per femtosecond, and state precisely what an experiment watching the atom for a year would see change.

Step 1. Rate: $\omega = \dfrac{|E|}{\hbar} = \dfrac{13.6\times1.602\times10^{-19}}{1.055\times10^{-34}} = \boxed{2.07\times10^{16}\ \mathrm{rad/s}}$ — twenty times faster than the $1\ \mathrm{eV}$ reference clock of the QM0 twin's worked example, because energy is rotation rate.

Step 2. Turns per femtosecond: $\dfrac{\omega\times10^{-15}}{2\pi} = \boxed{3.3\ \mathrm{turns/fs}}$. Over a year ($3.2\times10^7$ s) the dial completes $\sim10^{32}$ revolutions; over the age of the universe, $\sim10^{33}$ per few hundred years of that — an astronomically busy piece of mathematics.

Step 3. What changes observably: $\boxed{\text{nothing}}$. Because the state is stationary, every probability distribution — position, momentum, angular momentum, or any other observable — is identical at every instant. The sign of $E$ never enters any modulus. The clock becomes physical only in comparison: put the atom's electron in a superposition of $1s$ and $2p$ and the two clocks' rate difference $(E_{2p} - E_{1s})/\hbar$ beats at exactly the Lyman-α frequency — the spectroscopy bridge, built properly in Spectral Decomposition of Time Evolution.

2 · A frozen plane and a live gap (exam level)

A three-level system has $\hat H = \mathrm{diag}(2.0,\ 2.0,\ 3.2)\ \mathrm{eV}$ — a degenerate doublet and one higher level. A probe observable has matrix element $A_{13} = 0.40$ between levels 1 and 3 (and generic other entries). (a) Show the state $|u\rangle = (|1\rangle + |2\rangle)/\sqrt2$ is stationary and compute what any observable does in it. (b) For $|v\rangle = (|1\rangle + |3\rangle)/\sqrt2$, find the beat frequency, period, and the oscillation amplitude of $\langle\hat A\rangle$.

Step 1. (a) $\hat H|u\rangle = \tfrac{1}{\sqrt2}(2.0|1\rangle + 2.0|2\rangle)\ \mathrm{eV} = 2.0\ \mathrm{eV}\,|u\rangle$ — an eigenket of the degenerate eigenvalue. By the degeneracy derivation, $|u(t)\rangle = e^{-i(2.0\,\mathrm{eV})t/\hbar}|u\rangle$: every observable's expectation and full distribution in $|u\rangle$ is constant — the frozen plane in action, confirmed numerically to $10^{-12}$ for a generic probe.

Step 2. (b) The gap: $E_3 - E_1 = 1.2\ \mathrm{eV}$. Beat rate $\omega = \dfrac{1.2\times1.602\times10^{-19}}{1.055\times10^{-34}} = \boxed{1.82\times10^{15}\ \mathrm{rad/s}}$, period $T = \dfrac{2\pi}{\omega} = \boxed{3.45\ \mathrm{fs}}$.

Step 3. Amplitude: the interference term contributes $2|c_1||c_3||A_{13}|\cos(\omega t + \varphi)$, and with $|c_1| = |c_3| = \tfrac{1}{\sqrt2}$ this is $2\cdot\tfrac12\cdot0.40 = \boxed{0.40}$ of swing about the mean (peak-to-trough $0.80$), verified numerically. The diagonal terms sit rigid underneath, exactly as the freeze theorem demands.

Step 4. Moral for exams: stationarity questions are gap questions. List the distinct energies present in the state. Every pair of components with $\Delta E \neq 0$ contributes one oscillation frequency $\Delta E/\hbar$, while pairs with $\Delta E = 0$ contribute only time-independent terms. Here $|u\rangle$ has one distinct energy (silence) and $|v\rangle$ has one live gap (one beat frequency, no more).

Builds on — Time-Dependent Schrodinger Equation (this chapter's law of motion) · The Time-Evolution Operator (the spectral $\hat U$ whose fixed points these are) · Stationary States and Separation of Variables (the wave-register twin: separation construction, frozen-statistics and sharp-energy derivations, and the two-state beat are ITS, recalled here and never re-derived) · Eigenvalues and Eigenstates · Hermitian Operators and Observables · Born Rule and Measurement Probabilities.

Leads to — Spectral Decomposition of Time Evolution (all motion as energy interference) · Constants of the Motion and Conservation Laws (frozen observables, the complementary notion) · Ehrenfest's Theorem, dynamics register (rates of change via commutators) · The Schrodinger Picture (where these fixed points live as still scaffolding).

References — Griffiths, Introduction to Quantum Mechanics, 3rd ed., §2.1 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.1 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. III · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 4.

Provenance — Separation of variables and the form $\Psi_n=\psi_n e^{-iE_n t/\hbar}$ follow Griffiths §2.1, Sakurai §2.1, Cohen-Tannoudji Ch. III. A verifier should reconfirm: (1) the separation argument yielding $\hat H\psi=E\psi$ and $f=e^{-iEt/\hbar}$; (2) cancellation of the phase in $|\Psi_n|^2$; (3) the commutator/Ehrenfest argument giving $\mathrm{d}\langle\hat A\rangle_n/\mathrm{d}t=0$; (4) the two-level result $\langle\hat A\rangle=a\cos\omega_{21}t$; (5) the free-particle current $j=\hbar k/m$. The value $\hbar=1.054571817\times10^{-34}\,\mathrm{J\,s}$ is the CODATA reduced Planck constant. The square-well revival time $T_{\mathrm{rev}}=4mL^2/(\pi\hbar)$ follows from $E_n=n^2\pi^2\hbar^2/(2mL^2)$. (Lane note: items 1, 2 and 4 are owned by the wave-register twin Stationary States and Separation of Variables and are recalled, not reproduced; the Ehrenfest commutator engine belongs to the dynamics-register Ehrenfest's Theorem; the revival time is developed in Spectral Decomposition of Time Evolution. This article's fresh contributions — the converse fixed-point theorem, the distribution-level freeze, and the degenerate-plane theorem — are verified in its own script.)