Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

Part VI, Chapter 5 | Page 1 of 3

The Adiabatic Theorem

Slowly varying Hamiltonians, dynamic phase, and geometric phase

The adiabatic theorem addresses a fundamental question: what happens when a quantum system's Hamiltonian changes slowly in time? Unlike the sudden perturbation limit (where the state has no time to adjust) or the perturbative regime (where the perturbation is small), the adiabatic regime concerns arbitrary changes that are simply made slowly enough. The remarkable answer is that the system "follows" its instantaneous eigenstate, acquiring both a dynamic phase and a geometric (Berry) phase.

Statement of the Adiabatic Theorem

Consider a Hamiltonian $\hat{H}(t)$ that depends on time through some parameters $\vec{R}(t)$. At each instant, the instantaneous eigenstates and energies are defined by:

$$\hat{H}(t)|n(t)\rangle = E_n(t)|n(t)\rangle$$

The Adiabatic Theorem: If the system starts in the $n$-th eigenstate at $t = 0$, and the Hamiltonian changes sufficiently slowly, then at a later time $t$ the system remains in the $n$-th instantaneous eigenstate (up to a phase factor):

$$\boxed{|\Psi(t)\rangle = e^{i\theta_n(t)}\, e^{i\gamma_n(t)}\, |n(t)\rangle}$$

where $\theta_n(t)$ is the dynamic phase and $\gamma_n(t)$ is the geometric (Berry) phase. The system does not make transitions to other eigenstates -- it "follows" the instantaneous eigenstate as the Hamiltonian evolves.

Quantifying "Slow": The Adiabaticity Condition

The adiabatic approximation is valid when the rate of change of the Hamiltonian is small compared to the energy gap. The precise condition is:

$$\left|\frac{\langle m(t)|\dot{\hat{H}}(t)|n(t)\rangle}{(E_n(t) - E_m(t))^2/\hbar}\right| \ll 1 \quad \text{for all } m \neq n$$

Equivalently, if $T$ is the characteristic timescale of the Hamiltonian's variation and $\Delta E$ is the minimum energy gap to the nearest level:

$$\boxed{T \gg \frac{\hbar}{\Delta E_{\min}}}$$

Physical interpretation:

  • The time for change must be much longer than $\hbar/\Delta E$, the "quantum response time"
  • If the gap $\Delta E$ is large, the system can follow even relatively fast changes
  • If the gap closes ($\Delta E \to 0$), the adiabatic condition is impossible to satisfy, and transitions become inevitable

The Dynamic Phase

The dynamic phase is the natural generalization of the time-independent phase $e^{-iE_n t/\hbar}$:

$$\boxed{\theta_n(t) = -\frac{1}{\hbar}\int_0^t E_n(t')\, dt'}$$

This is simply the accumulated phase from the time-varying energy. If $E_n$ were constant, this would reduce to $\theta_n = -E_n t/\hbar$, the familiar stationary-state phase. The dynamic phase is always present and depends on the rate at which the Hamiltonian is varied (not just the path in parameter space).

The Geometric (Berry) Phase

The Berry phase is a purely geometric contribution discovered by Michael Berry in 1984. For a Hamiltonian parameterized by $\vec{R}(t)$:

$$\gamma_n(t) = i\int_0^t \langle n(t')|\frac{\partial}{\partial t'}|n(t')\rangle\, dt'$$

For a cyclic evolution (parameters return to initial values: $\vec{R}(T) = \vec{R}(0)$), this becomes a line integral over the closed path $\mathcal{C}$ in parameter space:

$$\boxed{\gamma_n = i\oint_{\mathcal{C}} \langle n(\vec{R})|\vec{\nabla}_R|n(\vec{R})\rangle \cdot d\vec{R}}$$

Remarkable properties of the Berry phase:

  • Geometric: Depends only on the path $\mathcal{C}$ in parameter space, not on the rate of traversal
  • Gauge invariant: Independent of the (arbitrary) phase convention for $|n(\vec{R})\rangle$ (for closed paths)
  • Real: $\gamma_n$ is always real (since $\langle n|n\rangle = 1$ implies $\langle n|\dot{n}\rangle$ is purely imaginary)
  • Observable: Can be measured via interference experiments

Berry Connection and Berry Curvature

The Berry phase has a beautiful differential-geometric structure. Define the Berry connection (analogous to the vector potential in electromagnetism):

$$\vec{\mathcal{A}}_n(\vec{R}) = i\langle n(\vec{R})|\vec{\nabla}_R|n(\vec{R})\rangle$$

The Berry curvature (analogous to the magnetic field) is its curl:

$$\vec{\mathcal{F}}_n(\vec{R}) = \vec{\nabla}_R \times \vec{\mathcal{A}}_n(\vec{R})$$

By Stokes' theorem, the Berry phase for a closed loop equals the flux of the Berry curvature through any surface bounded by the loop:

$$\gamma_n = \oint_{\mathcal{C}} \vec{\mathcal{A}}_n \cdot d\vec{R} = \int_{\mathcal{S}} \vec{\mathcal{F}}_n \cdot d\vec{S}$$

This parallel with electromagnetism is deep: the Berry connection is a gauge field, the Berry curvature is a field strength, and the Berry phase is a holonomy. This connection has profound implications in condensed matter physics (topological insulators) and gauge field theory.

Example: Particle in an Expanding Box

An infinite square well of length $L(t)$ that expands slowly. The instantaneous eigenstates and energies are:

$$\psi_n(x,t) = \sqrt{\frac{2}{L(t)}}\sin\!\left(\frac{n\pi x}{L(t)}\right), \quad E_n(t) = \frac{n^2\pi^2\hbar^2}{2mL(t)^2}$$

If the expansion is adiabatic ($\dot{L}/L \ll E_1/\hbar$):

  • The quantum number $n$ remains constant (no transitions)
  • The energy decreases as $L^{-2}$ (doing work against the walls)
  • The wave function continuously adjusts to fit the instantaneous box size
  • The Berry phase vanishes for this one-parameter problem (no enclosed area in parameter space)

Looking Ahead

On the next page, we compute the Berry phase explicitly for a spin-1/2 particle in a rotating magnetic field -- the canonical example that reveals the geometric origin of the Berry phase as half the solid angle subtended by the field's path. We also explore the Aharonov-Bohm effect and applications to molecular and solid-state physics.

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