The Adiabatic Theorem

Derivation

The Adiabatic Theorem

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Statement

Let a system evolve under a Hamiltonian \(\hat H(t)\) whose instantaneous eigenstates satisfy \(\hat H(t)\,|n(t)\rangle = E_n(t)\,|n(t)\rangle\) with the target level \(m\) non-degenerate and separated from every other level by a nonzero gap. If the system starts in \(|m(0)\rangle\) and \(\hat H(t)\) changes slowly enough that \(\hbar\,|\langle n(t)|\dot H(t)|m(t)\rangle| \ll |E_n(t)-E_m(t)|^2\) for all \(n\neq m\), then at every later time the state remains, up to phase, the instantaneous eigenstate \(|m(t)\rangle\): \(|c_m(t)|^2 \to 1\). The retained phase is the sum of a dynamical part and a geometric (Berry) part.

Why it matters

The adiabatic theorem is the license behind an enormous amount of physics: it tells you when you may follow a state along a slowly deformed spectrum without worrying about excitations. It underpins the Born–Oppenheimer separation of nuclear and electronic motion, adiabatic quantum computation, molecular reaction dynamics, and the quantization of transport in slowly driven systems.

It also isolates exactly one dimensionless number — the ratio of the drive rate to the squared level gap — that decides whether a process is reversible in the quantum sense. That same ratio, read the other way, tells you how to deliberately drive transitions (Landau–Zener) when the gap is small.

Assumptions

Discrete, non-degenerate spectrum near level \(m\).If \(E_m\) is degenerate the naive theorem fails; one must diagonalise \(\dot H\) within the degenerate subspace (Wilczek–Zee), and the retained phase becomes a non-Abelian holonomy rather than a number.

A nonzero gap \(|E_n-E_m|>0\) that never closes during the evolution.If a gap closes, the adiabatic parameter diverges and population leaks resonantly into the crossing level; the theorem gives no protection at a genuine level crossing.

The Hamiltonian is smooth (at least once differentiable) in \(t\), and \(\dot H\) is bounded.Sudden jumps or non-analytic kinks in \(\hat H(t)\) inject high-frequency Fourier components that resonate with the gap and spoil the exponential suppression; the estimate \(\sim \hbar\langle n|\dot H|m\rangle/(E_n-E_m)^2\) only bounds transitions for slowly varying, differentiable driving.

Evolution is unitary and closed (no coupling to a continuum or bath).With a continuum of final states or dissipation there is no isolated gap to protect the level; adiabatic following must be reformulated (e.g. adiabatic elimination or open-system adiabatic theorems).

Derivation

\[ i\hbar\,\frac{\partial}{\partial t}\,|\psi(t)\rangle = \hat H(t)\,|\psi(t)\rangle, \qquad \hat H(t)\,|n(t)\rangle = E_n(t)\,|n(t)\rangle,\quad \langle n(t)|m(t)\rangle = \delta_{nm}. \]

The exact Schrödinger equation together with the instantaneous eigenproblem solved at each fixed \(t\); the eigenstates form a complete orthonormal basis at every instant. A

\[ |\psi(t)\rangle = \sum_n c_n(t)\, e^{\,i\theta_n(t)}\,|n(t)\rangle, \qquad \theta_n(t) \equiv -\frac{1}{\hbar}\int_0^t E_n(t')\,dt'. \]

Expand in the instantaneous basis and factor out the dynamical phase \(\theta_n\) by hand. This is an exact change of variables, not an approximation; it is chosen so the leading energy term cancels in the next steps. B

\[ i\hbar\sum_n\!\Big[\dot c_n + i\dot\theta_n c_n\Big]e^{i\theta_n}|n\rangle + i\hbar\sum_n c_n e^{i\theta_n}|\dot n\rangle = \sum_n c_n e^{i\theta_n} E_n |n\rangle. \]

Insert the ansatz into step 1 and apply the product rule to \(\big(c_n e^{i\theta_n}|n\rangle\big)\); the right side used \(\hat H|n\rangle=E_n|n\rangle\). Here \(|\dot n\rangle \equiv \frac{d}{dt}|n(t)\rangle\). B

\[ i\hbar\,\dot\theta_n = i\hbar\!\left(-\frac{E_n}{\hbar}\right) = -iE_n \;\Rightarrow\; i\hbar(i\dot\theta_n c_n) = E_n c_n \;\Rightarrow\; \sum_n\!\Big[i\hbar\,\dot c_n\,|n\rangle + i\hbar\, c_n|\dot n\rangle\Big]e^{i\theta_n} = 0. \]

By construction \(\dot\theta_n=-E_n/\hbar\), so the \(i\hbar\,i\dot\theta_n c_n\) term exactly cancels the \(E_n c_n\) term on the right. Only the \(\dot c_n\) and \(|\dot n\rangle\) terms survive. B

\[ \dot c_m = -\sum_n c_n\, e^{\,i(\theta_n-\theta_m)}\,\langle m|\dot n\rangle. \]

Project step 4 onto \(\langle m|\), use \(\langle m|n\rangle=\delta_{mn}\), and divide by \(i\hbar\) (which cancels). This is still exact: it is the Schrödinger equation rewritten as coupled equations for the amplitudes. B

\[ \dot c_m = -\,c_m\,\langle m|\dot m\rangle \;-\; \sum_{n\neq m} c_n\, e^{\,i(\theta_n-\theta_m)}\,\langle m|\dot n\rangle. \]

Split the sum into the diagonal self-term (\(n=m\)) and the off-diagonal coupling (\(n\neq m\)). The diagonal term will generate the geometric phase; the off-diagonal terms are the transitions we must bound. B

\[ \frac{d}{dt}\big(\hat H|n\rangle\big):\;\; \dot H|n\rangle + \hat H|\dot n\rangle = \dot E_n|n\rangle + E_n|\dot n\rangle \;\;\xrightarrow{\;\langle m|,\, m\neq n\;}\;\; \langle m|\dot n\rangle = \frac{\langle m|\dot H|n\rangle}{E_n - E_m}. \]

Differentiate the eigenvalue equation and project onto \(\langle m|\) with \(m\neq n\); the \(\dot E_n\) term drops (\(\langle m|n\rangle=0\)) and \(\langle m|\hat H = E_m\langle m|\). This Feynman–Hellmann-type identity converts the geometric coupling \(\langle m|\dot n\rangle\) into a matrix element of \(\dot H\) divided by the gap — the object that must stay small. C

\[ \text{Drop } \sum_{n\neq m}\!\big(\cdots\big):\quad \dot c_m = -c_m\,\langle m|\dot m\rangle \;\Rightarrow\; c_m(t) = c_m(0)\,\exp\!\Big[-\!\int_0^t \langle m|\dot m\rangle\,dt'\Big] = c_m(0)\,e^{\,i\gamma_m(t)}. \]

The adiabatic approximation: neglect the off-diagonal terms (justified in step 9). What remains integrates exactly. Because \(\frac{d}{dt}\langle m|m\rangle=0\) forces \(\langle m|\dot m\rangle\) to be purely imaginary, \(\gamma_m(t)=i\!\int_0^t\langle m|\dot m\rangle\,dt'\) is real: the geometric (Berry) phase. Hence \(|c_m(t)|=|c_m(0)|\) — the occupation of level \(m\) is preserved. C

\[ \dot c_{n\neq m} \approx -\,e^{\,i(\theta_m-\theta_n)}\,\frac{\langle n|\dot H|m\rangle}{E_m-E_n}, \qquad |c_{n}(t)| \;\lesssim\; \frac{\hbar\,\big|\langle n|\dot H|m\rangle\big|}{(E_n-E_m)^2}. \]

Set \(c_m\approx 1\), \(c_{n\neq m}\approx 0\) on the right (first-order perturbation in the slowness). The phase \(\theta_m-\theta_n=-\frac1\hbar\!\int(E_m-E_n)dt'\) oscillates rapidly; integrating by parts against it produces the factor \(\hbar/(E_n-E_m)\), leaving the bound shown. The neglect in step 8 is legitimate precisely when this bound is \(\ll 1\). C

Result

\[ \boxed{\;\; \frac{\hbar\,\big|\langle n(t)|\dot H(t)|m(t)\rangle\big|}{\big(E_n(t)-E_m(t)\big)^2}\;\ll\;1 \quad (\forall\, n\neq m) \;\;\Longrightarrow\;\; |\psi(t)\rangle \simeq e^{i\theta_m(t)}\,e^{i\gamma_m(t)}\,|m(t)\rangle \;\;} \]

Reading. If the Hamiltonian is stirred slowly compared with the intrinsic oscillation timescale \(\hbar/|E_n-E_m|\) set by the gap, a system launched in eigenstate \(m\) is carried along the instantaneous eigenstate \(|m(t)\rangle\), acquiring only two phases: the dynamical phase \(\theta_m=-\frac1\hbar\!\int E_m\,dt'\) (the naive "energy times time") and the geometric phase \(\gamma_m=i\!\int\langle m|\dot m\rangle\,dt'\) that depends only on the path traced in parameter space, not on how fast it is traversed.

Units check. \(\hbar\) has units \(\mathrm{J\cdot s}\); \(\dot H\) has units \(\mathrm{J/s}\); their product is \(\mathrm{J^2}\). The denominator \((E_n-E_m)^2\) is also \(\mathrm{J^2}\). The adiabatic parameter is therefore dimensionless, as a condition of the form "\(\ll 1\)" must be.

Limiting cases

Infinitely slow drive (\(\dot H\to 0\)): the parameter \(\to 0\); following is perfect and only the geometric phase survives around a closed loop.
Static Hamiltonian (\(\dot H=0\)): \(\gamma_m=0\), \(\theta_m=-E_m t/\hbar\); recovers ordinary stationary-state evolution \(e^{-iE_m t/\hbar}|m\rangle\).
Large gap (\(|E_n-E_m|\to\infty\)): parameter \(\to 0\) even for fast driving — well-separated levels are hard to excite.
Vanishing gap (avoided crossing, \(E_n-E_m\to\Delta_{\min}\)): the parameter peaks at the closest approach; this is exactly where Landau–Zener transitions occur.
Closed loop in parameter space: \(\theta_m\) is path-length dependent, but \(\gamma_m\) becomes the gauge-invariant Berry phase \(\oint \mathbf{A}_m\cdot d\mathbf{R}\).

Breaks when

Level crossing / vanishing gap. When \(E_n(t)\to E_m(t)\) the denominator collapses, the adiabatic parameter diverges, and population transfers resonantly. The theorem offers no protection; the correct local description is Landau–Zener, giving a finite jump probability \(P\approx e^{-2\pi\Gamma}\) with \(\Gamma\propto \Delta^2/\hbar|\dot E|\).
Fast or non-smooth driving. If \(\hat H(t)\) changes on a timescale comparable to or shorter than \(\hbar/|E_n-E_m|\), or contains jumps/kinks, high-frequency components resonate with the gap; the sudden approximation (state frozen in the old basis) replaces the adiabatic one.
Degenerate target level. With \(E_m\) degenerate, "the eigenstate" is ambiguous; the single-state ansatz of step 2 is invalid and one must track the whole degenerate subspace with a non-Abelian connection.
Continuum / open systems. When the level is embedded in or coupled to a continuum, there is no isolated gap; decay and dephasing overwhelm adiabatic following.

Failure modes

Confusing "slow" with "slow in seconds". Adiabaticity is set by the drive rate relative to the gap \(\hbar/|E_n-E_m|\), not by any absolute clock; a microsecond ramp can be sudden for a small gap and adiabatic for a large one.
Dropping the geometric phase. Students often keep only \(\theta_m\) and set \(\gamma_m=0\), silently choosing a gauge; for cyclic evolution \(\gamma_m\) is physical and gauge-invariant and must not be discarded.
Forgetting to re-solve the eigenproblem at each instant. Using the initial eigenstates \(|n(0)\rangle\) throughout instead of \(|n(t)\rangle\) mixes the adiabatic and sudden pictures and gives wrong amplitudes.
Applying the theorem through a crossing. Extending "the system stays in level \(m\)" across an avoided crossing predicts the opposite diabatic outcome when the gap is small.
Assuming \(\langle m|\dot m\rangle\) is real. It is purely imaginary; treating it as real makes \(\gamma_m\) a decay rate and violates normalisation.

Discussion

The physical content of the theorem is a competition of two timescales. The Hamiltonian imposes an external timescale \(\tau_{\text{ext}}\sim H/|\dot H|\) over which the system's parameters change. The spectrum imposes an internal timescale \(\tau_{\text{int}}\sim \hbar/|E_n-E_m|\), the period of the phase beating between levels \(n\) and \(m\). Adiabaticity is simply \(\tau_{\text{int}}\ll\tau_{\text{ext}}\): the internal dynamics "keeps up" and continuously reorganises the state to remain an eigenstate. The rapidly oscillating phase \(e^{i(\theta_n-\theta_m)}\) in step 9 is the mathematical expression of this — its fast rotation averages the off-diagonal driving to nearly zero.

The two surviving phases have completely different characters. The dynamical phase \(\theta_m\) is extensive in time and depends on how long you take; the geometric phase \(\gamma_m\) is a property of the trajectory in parameter space and is invariant under reparametrising the speed. For a closed loop \(\gamma_m\) becomes Berry's phase, the flux of a curvature (the Berry curvature) through the enclosed surface — a genuinely observable, gauge-invariant quantity measured in interference, in the integer quantum Hall effect, and in molecular spectroscopy through the geometric (molecular Aharonov–Bohm) phase.

A subtlety the elementary statement hides: the first-order bound of step 9 is not tight. A more careful treatment (integration by parts iterated, or the superadiabatic / Berry expansion) shows that for an analytic Hamiltonian the transition amplitude is exponentially small, \(|c_n|\sim e^{-\tau_{\text{ext}}/\tau_{\text{int}}}\), controlled by the distance to the nearest complex-time degeneracy (a singularity of \(E_n(t)-E_m(t)\) analytically continued). This is why adiabatic following is so remarkably robust in practice — far better than the power-law bound suggests — and it is also why non-smooth driving is so damaging: it destroys the analyticity that produces the exponential.

Common misconceptions. "Adiabatic" here has nothing to do with thermodynamic heat exchange; it means only that the change is slow relative to the quantum gap. And "stays in the eigenstate" does not mean the state is unchanged — the eigenstate \(|m(t)\rangle\) itself deforms continuously; what is preserved is the occupation \(|c_m|^2\), not the ket.

Worked examples

\[ \textbf{Spin-}\tfrac12\textbf{ in a slowly rotating field.}\quad \hat H(t) = -\tfrac12\hbar\,\omega_0\,\hat{\boldsymbol\sigma}\cdot\hat{\mathbf n}(t), \quad \omega_0 = \frac{g\mu_B B}{\hbar}. \]

A magnetic moment in a field of fixed magnitude \(B\) whose direction \(\hat{\mathbf n}(t)\) precesses about \(z\) at rate \(\omega\). The gap is the Larmor splitting \(E_+-E_-=\hbar\omega_0\); the drive rate is \(\omega\). B

\[ \text{Adiabatic parameter} \;\sim\; \frac{\hbar\,|\langle -|\dot H|+\rangle|}{(E_+-E_-)^2} \;\sim\; \frac{\hbar\,(\tfrac12\hbar\omega_0\,\omega)}{(\hbar\omega_0)^2} = \frac{\omega}{2\omega_0}. \]

\(|\dot H|\sim \tfrac12\hbar\omega_0\,\omega\) since only the direction rotates (\(|\dot{\hat n}|=\omega\)); the off-diagonal element is order \(\tfrac12\hbar\omega_0\). Adiabaticity requires \(\omega\ll\omega_0\). B

\[ B = 1\ \mathrm{T},\; g\approx 2,\; \mu_B = 9.274\times10^{-24}\ \mathrm{J/T} \;\Rightarrow\; \omega_0 = \frac{2\,(9.274\times10^{-24})(1)}{1.055\times10^{-34}} = 1.76\times10^{11}\ \mathrm{rad/s}. \]

Electron Larmor frequency at 1 T. A

\[ \omega = 1.0\times10^{6}\ \mathrm{rad/s} \;\Rightarrow\; \frac{\omega}{2\omega_0} = \frac{10^{6}}{2(1.76\times10^{11})} = 2.8\times10^{-6}. \]

Insert the chosen rotation rate. A

\[ \frac{\omega}{2\omega_0} \approx 2.8\times10^{-6} \ll 1 \;\Rightarrow\; \text{spin follows the field; flip probability } P \sim \Big(\tfrac{\omega}{2\omega_0}\Big)^2 \sim 8\times10^{-12}. \]

Reading. The spin remains aligned with the instantaneous field to nine significant figures; the tiny residue is the non-adiabatic leakage. Around one full cone it also picks up the Berry phase \(\gamma_\pm=\mp\tfrac12\Omega\), half the solid angle \(\Omega\) subtended by \(\hat{\mathbf n}\).

\[ \textbf{Electron in a slowly expanding infinite well.}\quad E_n(t) = \frac{n^2\pi^2\hbar^2}{2mL(t)^2}, \quad \dot L = \text{const}. \]

A 1D box whose wall moves outward at constant speed \(\dot L\). Compare the internal gap timescale with the external expansion timescale for the ground state (\(m=1\)); the nearest level is \(n=2\). B

\[ \tau_{\text{int}} = \frac{\hbar}{E_2-E_1} = \frac{\hbar}{3E_1}, \qquad \tau_{\text{ext}} = \frac{L}{\dot L}; \qquad \text{adiabatic} \iff \frac{\tau_{\text{int}}}{\tau_{\text{ext}}} = \frac{\hbar\,\dot L}{3E_1 L}\ll 1. \]

\(E_2-E_1 = 3E_1\). The ratio of timescales is the relevant dimensionless slowness parameter (equivalent to the boxed condition up to an order-unity matrix element). B

\[ L_0 = 1.0\ \mathrm{nm}:\quad E_1 = \frac{\pi^2(1.055\times10^{-34})^2}{2(9.11\times10^{-31})(1.0\times10^{-9})^2} = 6.0\times10^{-20}\ \mathrm{J} \approx 0.38\ \mathrm{eV}. \]

Ground-state energy at the initial width. Then \(E_2-E_1=3E_1=1.8\times10^{-19}\ \mathrm{J}\). A

\[ \dot L = 1.0\ \mathrm{m/s}:\quad \tau_{\text{int}} = \frac{1.055\times10^{-34}}{1.8\times10^{-19}} = 5.8\times10^{-16}\ \mathrm{s}, \qquad \tau_{\text{ext}} = \frac{10^{-9}}{1.0} = 1.0\times10^{-9}\ \mathrm{s}. \]

Evaluate both timescales with the chosen expansion speed. A

\[ \frac{\tau_{\text{int}}}{\tau_{\text{ext}}} = \frac{5.8\times10^{-16}}{1.0\times10^{-9}} = 5.8\times10^{-7} \ll 1 \;\Rightarrow\; \text{electron stays in the (widening) ground state.} \]

Reading. Even at 1 m/s the wall moves slowly on the quantum clock, so the electron's ground-state occupation is preserved while \(E_1\) drops as \(1/L^2\). To force excitation you would need a wall speed comparable to \(3E_1 L/\hbar \approx 1.7\times10^{6}\ \mathrm{m/s}\).

Problems

Show that \(\langle m|\dot m\rangle\) is purely imaginary, and hence that \(\gamma_m=i\!\int\langle m|\dot m\rangle\,dt'\) is real.
Solution
Normalisation gives \(\langle m|m\rangle=1\) for all \(t\). Differentiate: \(\frac{d}{dt}\langle m|m\rangle = \langle\dot m|m\rangle+\langle m|\dot m\rangle=0\). Since \(\langle\dot m|m\rangle=\langle m|\dot m\rangle^{*}\), we get \(\langle m|\dot m\rangle^{*}+\langle m|\dot m\rangle=0\), i.e. \(2\,\mathrm{Re}\langle m|\dot m\rangle=0\), so \(\langle m|\dot m\rangle\) is purely imaginary: \(\langle m|\dot m\rangle=i\beta(t)\) with \(\beta\) real. Then \(\gamma_m=i\!\int i\beta\,dt' = -\!\int\beta\,dt'\), real.
A field of magnitude 0.10 T rotates at \(\omega=5.0\times10^{7}\ \mathrm{rad/s}\). Is the electron spin evolution adiabatic? Compute \(\omega/(2\omega_0)\).
Solution
\(\omega_0=g\mu_B B/\hbar = 2(9.274\times10^{-24})(0.10)/(1.055\times10^{-34}) = 1.76\times10^{10}\ \mathrm{rad/s}\). Then \(\omega/(2\omega_0)=5.0\times10^{7}/(2\times1.76\times10^{10})=1.4\times10^{-3}\ll1\). Adiabatic; approximate flip probability \(\sim(1.4\times10^{-3})^2\approx2\times10^{-6}\).
For the expanding well, at what constant wall speed \(\dot L\) does the timescale ratio \(\tau_{\text{int}}/\tau_{\text{ext}}\) reach 1 for an electron with \(L=1.0\ \mathrm{nm}\)? Interpret.
Solution
Set \(\hbar\dot L/(3E_1 L)=1\Rightarrow \dot L = 3E_1 L/\hbar\). With \(E_1=6.0\times10^{-20}\ \mathrm J\), \(L=10^{-9}\ \mathrm m\): \(\dot L = 3(6.0\times10^{-20})(10^{-9})/(1.055\times10^{-34}) = 1.7\times10^{6}\ \mathrm{m/s}\). At this speed the wall moves as fast as the internal beat period allows, so the evolution crosses from adiabatic toward sudden and the ground state can no longer be tracked. Physically \(\dot L\) is a substantial fraction of a percent of \(c\) — expansion is adiabatic for any laboratory-scale mechanical speed.
Near an avoided crossing the two levels are \(E_\pm=\pm\tfrac12\sqrt{\Delta^2+(\alpha t)^2}\), with gap \(\Delta\) at \(t=0\) and sweep rate \(\alpha\). Estimate the adiabatic parameter at \(t=0\) and state the condition for adiabatic following.
Solution
At \(t=0\): \(E_+-E_-=\Delta\). \(\dot H\) has off-diagonal scale set by \(\frac{d}{dt}(\alpha t)=\alpha\), so \(|\langle-|\dot H|+\rangle|\sim \alpha/2\). Parameter \(\sim \hbar(\alpha/2)/\Delta^2\). Adiabatic following requires \(\hbar\alpha \ll \Delta^2\); the Landau–Zener jump probability is \(P=\exp(-\pi\Delta^2/2\hbar\alpha)\), which \(\to 0\) exactly in this limit and \(\to 1\) (fully diabatic) when \(\hbar\alpha\gg\Delta^2\).
A quantum harmonic oscillator has its frequency ramped slowly from \(\omega_i\) to \(\omega_f\) while the system sits in the ground state. Using the adiabatic theorem, what is the final state and its energy? What sets the slowness requirement?
Solution
The ground state is non-degenerate with gap \(\hbar\omega(t)\) to the first excited state. If \(|\dot\omega/\omega|\ll\omega\) (fractional change per oscillation period small), the system follows the instantaneous ground state \(|0(t)\rangle\). The final state is the ground state of the oscillator at \(\omega_f\), with energy \(E=\tfrac12\hbar\omega_f\) — not \(\tfrac12\hbar\omega_i\). The occupation \(|c_0|^2=1\) is preserved even though the energy changes, because the eigenstate itself deforms (the wavefunction width scales as \((\hbar/m\omega)^{1/2}\)). The slowness parameter is \(\sim|\dot\omega|/\omega^2\ll1\).