The Sudden Approximation

The sudden approximation is a standard theoretical framework for describing quantum quenches and rapid state preparation in cold-atom experiments: a "quench" — a parameter switched faster than $\hbar/\Delta E$ — is the opening move of experiments on thermalization, many-body localization, and dynamical phase transitions, and, when the switching time satisfies $\tau\ll\hbar/\Delta E$, the frozen-overlap rule provides the initial state immediately after the quench. Trap engineers run the other direction along our bridge curve: shortcut-to-adiabaticity protocols shape $\omega(t)$ so that a fast ramp lands on the adiabatic result, exploiting exactly the Ermakov dynamics integrated above. The precision frontier belongs to the next article's physics: KATRIN must model the sudden shake-up of the daughter ion in tritium decay to interpret its endpoint spectrum, and attosecond spectroscopy has begun to time suddenness itself.

Picture a marble at the bottom of a bowl. Slide the bowl slowly and the marble rides along, always at the bottom — the adiabatic story, told in The Adiabatic Theorem. Yank the bowl in one snap instead and the marble is left hanging where it was, now on the slope of the relocated bowl, and it starts to slosh. "Left hanging" is the frozen state vector; the sloshing is what a state that is no longer an eigenstate does under the new Hamiltonian. The key physical input is the first-order Schrödinger equation, which guarantees that the state remains continuous during a finite-time switch. The Schrödinger equation is first order in time, so $|\psi(t)\rangle$ cannot jump even when $H(t)$ does; over a time interval $\tau$, the state changes by an amount of order $\Delta E\,\tau/\hbar$, where $\Delta E$ is the energy uncertainty of the initial state, and when that is negligible the state emerges untouched. Everything else is wardrobe: the same ket, in a new energy basis, is suddenly a superposition. The doubled box is this article's exact laboratory, and its answer is worth staring at before any algebra. Yank the right wall from $L$ to $2L$: the frozen half-sine hump is not mostly the new ground state. The biggest single share — exactly one half — goes to the first excited state, whose full-sine shape matches the frozen hump on $[0,L]$ perfectly. The new ground state gets only $36\%$, and every even state above $n=2$ gets exactly nothing.

The second picture is the payoff: what happens between the extremes. Ramp an oscillator's frequency from $\omega_1$ to $5\omega_1$ over a duration $\tau$. For a harmonic oscillator with a time-dependent frequency, the Schrödinger equation can be solved exactly, allowing the entire crossover from the sudden to the adiabatic regime to be computed — a sudden plateau at $2\sqrt{5}/6 = 0.745$, an adiabatic climb to $1$, and a knee right where the criterion says, at $\omega_1\tau \sim 1$.

Split time into three acts: before ($H = H_i$, the system in an eigenstate), the switch ($H(t)$ interpolates from $H_i$ to $H_f$ over a duration $\tau$, in any manner whatsoever), and after ($H = H_f$, constant). The sudden approximation is a statement about act two. The evolution across the switch is the time-ordered exponential of Dyson Series and the Time-Ordered Evolution Operator, and its deviation from the identity is set by the time integral of the Hamiltonian:

where: $U(\tau,0)$ — the evolution operator across the switch; $\mathcal{T}$ — the time-ordering symbol; $H(t')$ — the interpolating Hamiltonian, bounded but otherwise arbitrary. A bounded generator acting for a vanishing time produces a vanishing rotation; the magnitude of $H_f - H_i$ never enters, only the product of energy scale and duration. Because $U \to \mathbb{1}$, the state is continuous across the switch: $|\psi(0^+)\rangle = |\psi(0^-)\rangle$. But the labels have changed. To say anything about energies you must expand the frozen ket over the new eigenstates, and the coefficients are pure geometry:

where: $|\phi_n^{(f)}\rangle$ — eigenstates of $H_f$; $|\psi(0)\rangle$ — the frozen state. The sum rule is automatic completeness: the new eigenstates form a basis, so squared overlaps of a unit vector sum to one. If the post-switch Hamiltonian is time independent, each component subsequently accumulates only a phase $e^{-iE_n^{(f)}t/\hbar}$, so the occupation probabilities remain constant. One quieter subtlety: energy. Whoever moved the Hamiltonian may have done work, so $\langle H_f \rangle \neq \langle H_i\rangle$ in general. But $\sum_n P_{i\to n} E_n^{(f)}$ is computable from the ladder, and in special geometries (the doubled box, where the state never touches the moved wall) no work is done and the mean energy is exactly conserved.

Take the ground state of the well of width $L$ — the spectrum $E_n = n^2\pi^2\hbar^2/2mL^2$ built in Infinite Square Well — and, at $t=0$, teleport the right wall to $2L$. The frozen wavefunction is the old half-sine hump on $[0,L]$, zero beyond. Both geometries have exact eigenstates, so every sudden amplitude is a closed-form integral. One rung deserves notice: the new $n=2$ state is $\sqrt{1/L}\,\sin(\pi x/L)$ — the same shape as the frozen state itself on $[0,L]$.

The moral survives beyond boxes: sudden processes are ruled by shape matching, not energy ordering. The state you land in most is the one that looks like you, and symmetry can silence whole families of destinations. The selection lemma deserves one clean statement: if the jump $H_i \to H_f$ leaves a symmetry intact (for a nuclear charge jump, full rotational symmetry), the frozen state's quantum numbers under that symmetry cannot change — the overlap of states in different symmetry sectors vanishes identically. Sudden switches conserve every quantum number the switch respects; that lemma kills the even rungs here and the entire $\ell \neq 0$ sector in the shake-up article next door.

The box laboratory is exact only at $\tau = 0$. To watch the sudden approximation fail gracefully into the adiabatic theorem we need dynamics exact at every ramp speed, and the harmonic oscillator delivers: a Gaussian evolving under $H = p^2/2m + \tfrac12 m\omega(t)^2 x^2$ remains Gaussian for any $\omega(t)$. The ground state of Quantum Harmonic Oscillator: Analytic Solution is a pure Gaussian, so the whole quench problem collapses to one ordinary differential equation.

Constant Perturbation Switched On studies what looks like the same experiment — $V$ appears abruptly at $t=0$ and stays — and finds populations that oscillate: $P_{i\to f}(t) = 4|V_{fi}|^2\sin^2(\omega_{fi}t/2)/(\hbar\omega_{fi})^2$. This article claims populations frozen forever. Both are right. The sudden rule counts populations in the new basis (eigenstates of $H_0 + V$): constant, and for weak $V$ equal to $|V_{fi}/(E_f - E_i)|^2$. The switched-on result predicts an oscillating transition probability in the old basis, whose long-time average is exactly twice the sudden-transition probability — the cross term of the two dressed components. Numerically, at $V/\Delta = 0.02$: new-basis $3.995\times10^{-4} \approx (V/\Delta)^2$, old-basis time average $2.004\times$ that. Same physics, two wardrobes; the deferral in that article — "the sudden and adiabatic extremes of switch-on are The Sudden Approximation and The Adiabatic Theorem" — lands here.

Example 1: the doubled box, by the numbers

A particle sits in the ground state of a well of width $L$; the right wall jumps to $2L$ instantaneously. Where does it land?

Step 1. Enumerate destinations with $c_n = \tfrac{4\sqrt2}{\pi}\tfrac{\sin(n\pi/2)}{4-n^2}$ and $c_2 = 1/\sqrt2$: $P_1 = 32/9\pi^2 = \mathbf{0.3603}$, $P_2 = \mathbf{0.5000}$ (exact), $P_3 = 0.1297$, $P_4 = 0$, $P_5 = 0.00735$, $P_6 = 0$, $P_7 = 0.00160$.

Step 2. Check completeness: the seven rungs already carry $0.9990$; summing to $n = 2\times10^4$ gives $\sum P_n = \mathbf{1.0000000}$.

Step 3. Check the energy ledger: $\langle E\rangle_{\text{after}} = E_1^{(L)}\cdot\tfrac14\sum P_n n^2 = \mathbf{E_1^{(L)}}$ — exactly conserved, because the frozen state never touches the moved wall.

Step 4. Read the physics: the single most likely outcome is the first excited state, at exactly $50\%$ — shape matching beats energy proximity.

Example 2: how fast must a trap quench be?

An ion oscillates at $\omega_1/2\pi = 10\,$MHz; the trap is ramped to $\omega_2 = 2\omega_1$. When is the ramp sudden?

Step 1. The clock is $1/\omega_1 = \mathbf{15.9\ \text{ns}}$.

Step 2. A $\tau = 1\,$ns ramp has $\omega_1\tau = 0.063$: the exact bridge gives $P(0\to 0) = 0.9428$, indistinguishable from the sudden overlap $2\sqrt2/3 = \mathbf{0.9428}$.

Step 3. A $\tau = 100\,$ns ramp has $\omega_1\tau = 6.3$: the bridge gives $P = \mathbf{0.997}$ — mostly adiabatic; the sudden estimate ($0.943$) is badly off.

Step 4. A $\tau = 1\,\mu$s ramp gives $P = \mathbf{1.0000}$: fully adiabatic. The design space of quench experiments lives on this one curve, with the crossover at the $\tau \sim 1/\omega_1$ knee the variance bound predicts.

References. Griffiths, 3rd ed., §11.5 · Sakurai & Napolitano §5.6 · Shankar, Ch. 18 · Cohen-Tannoudji, Diu & Laloë, Vol. II · Messiah, Quantum Mechanics, Vol. II, Ch. XVII (the variance bound) · Güttinger, Z. Phys. 73, 169 (1932).

Note on a circulating error. An earlier version of this material quoted the tritium ground-state survival probability as $P_{1s\to1s}=2^9(ZZ')^3/(Z+Z')^6=4096/4913\approx0.834$. That expression fails three ways: the coefficient $2^9$ should be $2^6$ (it is the amplitude $8(ZZ')^{3/2}/(Z+Z')^3$ that gets squared, using $\int_0^\infty r^2 e^{-\beta r}\,dr=2/\beta^3$); the fraction $4096/4913$ has denominator $17^3$ with no 17 anywhere in the physics; and $0.834$ is (approximately) the amplitude $0.838$, not the probability. The correct value is $P = 512/729 = 0.702$, derived in Beta Decay and Atomic Shake-Up as Sudden Processes. The well-doubling value $32/9\pi^2 = 0.360$ is correct.

Builds on: The Interaction Picture and Time-Dependent Hamiltonians · Time-Dependent Perturbation Theory: General Formalism · First-Order Transition Amplitudes and Probabilities. Leads to: Constant Perturbation Switched On · Fermi's Golden Rule · Density of Final States — and, in this band, Beta Decay and Atomic Shake-Up as Sudden Processes and Landau–Zener Transitions.