Adiabatic Theorem

Statement

If a Hamiltonian $H(t)$ changes slowly and an instantaneous eigenvalue $E_n(t)$ remains isolated by a gap, a state initially in the eigenstate $\lvert n(0)\rangle$ remains close to the corresponding instantaneous eigenstate:

$$\lvert\psi(t)\rangle \approx e^{i\gamma_n(t)} \exp \left( -\frac{i}{\hbar}\int_0^t E_n(t')\,dt' \right) \lvert n(t)\rangle.$$

Here $\gamma_n(t)$ is the geometric phase in a chosen gauge:

$$\gamma_n(t) = i\int_0^t \langle n(t')\vert\dot n(t')\rangle\,dt'.$$

Common Slowness Diagnostic

For a nondegenerate level, a useful diagnostic is

$$\max_{m\ne n} \frac{ \hbar\, \lvert\langle m(t)\vert\dot H(t)\vert n(t)\rangle\rvert }{ \lvert E_m(t)-E_n(t)\rvert^2 } \ll 1,$$

with the warning that rigorous adiabatic theorems require more precise hypotheses.

Assumptions

• The relevant eigenvalue or eigenspace remains separated from the rest of the spectrum.
• $H(t)$ is sufficiently smooth over the time interval.
• The initial state lies in the isolated eigenspace being followed.
• The total evolution time is long compared with inverse gap scales in the relevant dimensionless sense.
• Degenerate eigenspaces require a non-Abelian adiabatic connection rather than a single phase.

Quantum-Mechanical Meaning

The theorem explains why slow parameter changes can transport eigenstates without inducing transitions. It underlies adiabatic state preparation, Berry phase, molecular Born–Oppenheimer reasoning, and adiabatic quantum computation.

It is not a statement that the state has zero time dependence. The state accumulates a dynamical phase, may accumulate a geometric phase, and can fail near avoided crossings if the evolution is not slow compared with the squared gap scale.

Canonical Links

• Time-Dependent Hamiltonians
• Adiabatic Theorem Reminder
• Berry Phase
• Non-Abelian Berry Phase Preview
• Berry Connection as a Mathematical Object
• Small Parameters and Error Estimates
• Landau–Zener Transition
• Sudden Approximation
• Adiabatic Approximation as a Method
• STIRAP

Common Mistakes

• Saying “slow” without identifying a gap and a dimensionless slowness parameter.
• Applying the nondegenerate formula through a level crossing.
• Forgetting the Berry phase.
• Confusing adiabatic following with staying in a fixed initial vector.
• Assuming an adiabatic approximation is automatically accurate for arbitrarily long times without checking accumulated errors.

Quick Check

Why does a small energy gap make adiabatic following harder?

Solution

The transition diagnostic contains the squared gap in the denominator. If $\lvert E_m-E_n\rvert$ becomes small, the same rate of change $\dot H$ produces a larger transition amplitude. Near an actual crossing, the isolated-eigenstate assumption fails.

References

• M. Born and V. Fock, “Beweis des Adiabatensatzes,” Zeitschrift für Physik 51, 165-180, 1928.
• T. Kato, “On the adiabatic theorem of quantum mechanics,” Journal of the Physical Society of Japan 5, 435-439, 1950.
• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45-57, 1984.
• A. Messiah, Quantum Mechanics, Dover, 1999.