Stationary States and Energy

Every molecular orbital diagram drawn in an organic chemistry course, every energy level reported by a DFT or Hartree–Fock calculation, and every peak position in a UV-Vis, IR, or NMR spectrum is a statement about stationary states. In electronic structure software, computed orbital energies (Kohn–Sham eigenvalues, Hartree–Fock orbital energies) are meaningful precisely because a single Slater determinant built from occupied stationary-like orbitals gives a genuinely time-independent electron density in the mean-field approximation — the entire logical architecture of "an atom's electron configuration" depends on this. In spectroscopy, line positions in absorption and emission spectra are measured Bohr frequencies $\nu = (E_m-E_n)/h$ between pairs of stationary states, and the narrowness of a spectral line is a direct experimental signature of how close to perfectly stationary (how long-lived) the initial and final states are — natural linewidths in atomic clocks, for instance, are limited by the finite lifetime (finite "non-stationarity," strictly) of the excited clock state. In photochemistry and femtosecond spectroscopy, deliberately preparing coherent superpositions of vibrational or electronic stationary states creates "wavepackets" whose motion — bond stretching, bond breaking, conical-intersection dynamics — is entirely described by the beating cross terms derived above, oscillating at the relevant Bohr frequencies; pump–probe experiments literally watch these beats in real time. In quantum computing with molecular or atomic qubits, coherence times are set by how long a prepared superposition of two stationary states can avoid decohering back toward populating a single eigenstate, making the stability (or fragility) of superpositions relative to true stationary states a live engineering concern.

Ask the tutor: If a stationary state's probability density never changes, what physically distinguishes a ground-state atom from one sitting in an excited stationary state, if not "motion" of the electron cloud?

Start from the The Time-Dependent Schrodinger Equation,

and ask: for what class of wavefunctions is the time-dependence as simple as possible? Try separation of variables, writing $\Psi(\vec r, t) = \psi(\vec r)\,f(t)$, on the assumption that $\hat H$ has no explicit time-dependence (no oscillating field, no moving nuclei — just a fixed potential $V(\vec r)$). Substituting and dividing through by $\psi(\vec r) f(t)$ separates the equation into a purely spatial piece and a purely temporal piece, each equal to a constant that we call $E$. The spatial piece is exactly the The Time-Independent Schrodinger Equation, $\hat H \psi(\vec r) = E\psi(\vec r)$, and the temporal piece integrates immediately to $f(t) = e^{-iEt/\hbar}$. The crucial physical content is what happens to the modulus of this factor. Writing $E$ as a real number (guaranteed because $\hat H$ is Hermitian and $E$ is an expectation value of a physical, measurable energy), the complex exponential $e^{-iEt/\hbar}$ traces a point moving around the unit circle in the complex plane as $t$ increases — its real and imaginary parts oscillate, but its magnitude is always exactly 1. When you compute a probability density you take $|\Psi|^2 = \Psi^*\Psi$, and multiplying by a unit-modulus number and then by its complex conjugate always returns 1:

The exponential factors cancel identically, for every $\vec r$ and every $t$. This is not an approximation, not a slow decay, not a "quasi-stationary" statement — it is an algebraic identity. The probability density of an eigenstate of $\hat H$ is exactly constant in time. This has an immediate chemical translation: orbitals, as usually drawn (electron clouds, contour surfaces, nodal patterns), are pictures of $|\psi_n(\vec r)|^2$, and since that quantity never depends on $t$, the picture is legitimately a single, permanent snapshot rather than a frame from a movie. The electron is not "sitting still" — momentum, angular momentum, and the phase of $\Psi$ are all still dynamical — but everything an experiment can extract from $|\Psi|^2$ (position probabilities, and by extension anything built from position-diagonal operators) is frozen. It is essential to see why this fails the moment you leave a single eigenstate. Consider a superposition of two stationary states with different energies,

Now $|\Psi|^2$ picks up cross terms $c_1^*c_2\,\psi_1^*\psi_2\,e^{-i(E_2-E_1)t/\hbar} + \text{c.c.}$, and these do not cancel — they oscillate at the Bohr angular frequency $\omega_{21} = (E_2-E_1)/\hbar$. This is precisely the mechanism behind time-dependent probability densities, transition dipole oscillations, and spectroscopic line frequencies $\nu = (E_2-E_1)/h$. Stationary states are therefore the building blocks out of which all time-dependent chemical dynamics (bond vibration beating, electron transfer, wavepacket motion) is assembled, and understanding why a single eigenstate is inert is what lets you correctly identify why a superposition is not.

The definition of a stationary state:

where: $\hat H$ is the (time-independent) Hamiltonian operator, $\psi_n(\vec r)$ is a time-independent spatial eigenfunction, and $E_n$ is the real eigenvalue — a single sharp energy value associated with that eigenfunction.

Reading: this is the same equation as the The Time-Independent Schrodinger Equation — it says nothing yet about time, only that $\psi_n$ is a special function for which applying $\hat H$ just rescales it by $E_n$ rather than distorting its shape. The full time-dependent stationary-state wavefunction:

where: $\Psi_n(\vec r,t)$ solves the full The Time-Dependent Schrodinger Equation, $\psi_n(\vec r)$ is the same spatial eigenfunction as above, $E_n$ is its eigenvalue, $\hbar$ is the reduced Planck constant, and $i=\sqrt{-1}$.

Reading: all of the time-dependence of an energy eigenstate is packaged into one complex exponential multiplying a completely static spatial shape — the "shape" of the wavefunction never changes, only its overall phase angle rotates uniformly in time. The invariance of the probability density:

where: the left side is the quantum mechanical probability density at time $t$; the right side shows it equals the time-independent quantity $|\psi_n(\vec r)|^2$ for all $t$.

Reading: because $e^{-iE_nt/\hbar}\cdot e^{+iE_nt/\hbar} = 1$ identically, the "movie" of the electron's probability cloud is a single still frame played forever — this is the literal meaning of the word stationary. The invariance of expectation values of time-independent operators:

where: $\hat A$ is any observable operator (position, momentum, kinetic energy, dipole moment operator) that does not itself depend explicitly on $t$; $\langle \hat A\rangle_n$ is the expectation value in state $n$.

Reading: because the phase factor and its conjugate always cancel wherever $\hat A$ doesn't act on $t$ directly, every measurable average computed from a single stationary state is a fixed number — this is why "the dipole moment of the ground state" or "the average kinetic energy of an electron in a $2p$ orbital" are meaningful constants rather than time-labeled quantities. The Bohr frequency condition, showing what breaks the stationarity:

where: $\omega_{mn}$ is the angular frequency and $\nu_{mn}$ the ordinary frequency at which cross terms between states $m$ and $n$ oscillate in a superposition, and $h = 2\pi\hbar$ is Planck's constant.

Reading: a single stationary state has $E_m = E_n$ trivially (same state), so $\omega_{mn}=0$ — nothing oscillates; only genuine superpositions of different energies produce the nonzero beat frequency that spectroscopy measures as an absorption or emission line.

Example 1 — Foundational: the particle in a box, checking stationarity explicitly. Consider an electron confined to a one-dimensional box of length $L = 0.53\ \text{nm}$ (a rough proxy for a conjugated chain), in the $n=3$ eigenstate. The normalized eigenfunctions and eigenvalues are

Plugging in $h = 6.626\times10^{-34}\ \text{J s}$, $m_e = 9.109\times10^{-31}\ \text{kg}$, $L = 5.30\times10^{-10}\ \text{m}$:

Numerator: $9 \times 4.390\times10^{-67} = 3.951\times10^{-66}\ \text{J}^2\text{s}^2$. Denominator: $8 \times 9.109\times10^{-31}\times 2.809\times10^{-19} = 2.047\times10^{-48}\ \text{kg m}^2$.

The full time-dependent stationary state is $\Psi_3(x,t) = \psi_3(x)e^{-iE_3t/\hbar}$. Compute the probability density at $t=0$ and at a later, deliberately awkward time $t_1 = 3.7\times10^{-16}\ \text{s}$:

At $t_1$ the phase angle accumulated is

an enormous, essentially arbitrary angle on the unit circle — yet

identical to the $t=0$ result, term for term, regardless of how large or awkward $\phi$ is. Limiting-case check: as $t\to\infty$, $\phi\to\infty$ and the phase winds around the circle infinitely many times, yet $|\Psi_3|^2$ never changes at all — the density has no time-decay, no dephasing, no drift, confirming the exact cancellation is not merely a short-time approximation.

Example 2 — Applied: population of a single vibrational eigenstate and its dipole moment. A diatomic molecule (take CO, reduced mass $\mu = 6.860$ u $= 1.139\times10^{-26}\ \text{kg}$, force constant $k = 1902\ \text{N/m}$) sits purely in its harmonic-oscillator vibrational ground state $v=0$, i.e. genuinely a single stationary state with no admixture of $v=1$ or higher. Its angular frequency is

and its energy eigenvalue is

The full stationary wavefunction is $\Psi_0(x,t) = \psi_0(x)\,e^{-iE_0t/\hbar}$, where $\psi_0(x)$ is the real Gaussian ground-state harmonic-oscillator function. Because the vibrational dipole operator (to leading order, $\hat\mu(x) \approx \mu_0 + (d\mu/dx)_0 x$) is time-independent and this is a pure eigenstate, its expectation value is

since $\int \psi_0^* x\,\psi_0\,dx = 0$ by the symmetry of the ground-state Gaussian about $x=0$ — a constant, time-independent value equal simply to the permanent dipole moment $\mu_0$, unaffected by the phase factor. No IR emission or absorption occurs from this state alone: a pure vibrational eigenstate does not radiate, however long you wait. Limiting-case check: setting $(d\mu/dx)_0 \to$ any nonzero value changes nothing about this conclusion, because the integral — not the size of the dipole derivative — is what enforces zero net oscillating dipole in a single eigenstate; only mixing in $v=1$ (with its odd-parity wavefunction) makes $\int \psi_0^* x\,\psi_1\,dx \ne 0$ and produces a genuinely oscillating, IR-active dipole.

Example 3 — Exam/competition level: a two-level system with unequal, non-round coefficients and a degenerate twist. A model two-state system (e.g. a simplified treatment of an avoided crossing) has non-degenerate eigenstates $\psi_1,\psi_2$ with $E_1 = 2.35\ \text{eV}$ and $E_2 = 3.80\ \text{eV}$, prepared at $t=0$ in the superposition

already normalized since $0.60^2+0.800^2 = 0.36+0.64=1.00$. This is not stationary. The full time-dependent wavefunction is

The Bohr frequency governing oscillation of any cross-coupled observable is

corresponding to an oscillation period

If $\psi_1,\psi_2$ have a nonzero transition dipole matrix element $\langle\psi_1|\hat x|\psi_2\rangle = 0.045\ \text{nm}$ (a representative off-diagonal position matrix element), the expectation value of position oscillates as

a genuine, measurable oscillation with amplitude $0.0432\ \text{nm}$ and period $2.85\ \text{fs}$ — this is the mathematical seed of a Rabi-like coherent oscillation. Limiting-case check: if instead $E_2 \to E_1$ (degeneracy), $\omega_{21}\to 0$ and $\cos(\omega_{21}t+\theta_{12}) \to \cos(\theta_{12})$, a genuine constant — the oscillation freezes and the superposition becomes stationary again, exactly consistent with Step 6 of the second derivation above. Also check the opposite population extreme: setting the coefficient on $\psi_2$ to zero ($0.800\to 0$, forcing $0.60\to1$ for normalization) collapses the cross term's prefactor $2(1)(0)=0$ to zero identically, correctly recovering a pure, stationary $\psi_1$ eigenstate with no oscillation at all.

The concept of stationary states traces to Niels Bohr's 1913 postulate that atoms exist only in discrete "stationary states" with fixed energies, radiating only when jumping between them — a bold guess that predated any wave mechanics. It was Erwin Schrödinger who, in 1926, gave the idea rigorous mathematical footing: solving his newly written wave equation showed that the allowed energies of the hydrogen atom emerge naturally as eigenvalues, and that the corresponding eigenfunctions carry exactly the frozen-in-time probability structure that Bohr's stationary states had only been postulated to have. Schrödinger's paper series that year unified Bohr's phenomenological quantization with a genuine differential-equation framework, and it was in this same work that the time-dependent equation and its separation into spatial and temporal pieces — precisely the derivation above — first appeared in essentially modern form.