Adiabatic Theorem Reminder

Berry phase is usually introduced after saying that a Hamiltonian changes “slowly” and that the state stays in an instantaneous eigenstate. This page records exactly what that sentence is doing. The theorem statement belongs to the Adiabatic Theorem card, while runtime diagnostics and error estimates belong to Adiabatic Approximation as a Method. Here the goal is to isolate the ingredient needed for geometric phase.

Let a Hamiltonian depend on externally controlled parameters $R(t)$:

$$H(R(t))\lvert n(t)\rangle = E_n(t)\lvert n(t)\rangle.$$

The basic adiabatic statement is: if the state starts in an isolated instantaneous eigenspace and the Hamiltonian changes slowly compared with the relevant gap scales, then the state remains close to that instantaneous eigenspace. It does not remain equal to the same vector. It accumulates phase, and that phase has both dynamical and geometric parts.

Minimal Statement

For a nondegenerate eigenvalue $E_n(t)$ separated from the rest of the spectrum, adiabatic evolution gives

$$\lvert\psi(t)\rangle = e^{i\alpha_n(t)} \lvert n(t)\rangle + O(\epsilon)$$

when the initial state is $\lvert n(0)\rangle$ and the dimensionless adiabatic error $\epsilon$ is small. The phase is

$$\alpha_n(t) = -\frac{1}{\hbar} \int_0^t E_n(t')\,dt' + i\int_0^t \langle n(t')|\dot n(t')\rangle\,dt'.$$

The first term is the dynamical phase. The second term is the geometric phase accumulated along the path already traversed in parameter space. For a closed path, it becomes the Berry phase.

The formula assumes a chosen smooth phase convention for $\lvert n(t)\rangle$ along the path. For a closed loop, the final phase factor is independent of smooth single-valued gauge choices, modulo $2\pi$.

What Slow Means

For a nondegenerate finite-dimensional problem, a common diagnostic is

$$\epsilon_{mn}(t) = \frac{ \hbar\, \lvert\langle m(t)|\dot n(t)\rangle\rvert }{ \lvert E_m(t)-E_n(t)\rvert } \ll 1 \qquad (m\ne n).$$

Using the differentiated eigenvalue equation, this can be written as

$$\epsilon_{mn}(t) = \frac{ \hbar\, \lvert\langle m(t)|\dot H(t)|n(t)\rangle\rvert }{ \lvert E_m(t)-E_n(t)\rvert^2 } \ll 1$$

when $E_m(t)\ne E_n(t)$. This ratio is a diagnostic, not the whole theorem. Rigorous adiabatic results require assumptions about smoothness, domains for unbounded operators, endpoints, total time, and how the gap behaves over the interval.

The squared gap in the denominator is the practical lesson. A small gap makes adiabatic following difficult. A closing gap invalidates the isolated-eigenspace assumption.

Deriving the Phase Once Following Is Assumed

Suppose the adiabatic theorem has already justified the one-state ansatz

$$\lvert\psi(t)\rangle = e^{i\alpha_n(t)} \lvert n(t)\rangle.$$

Insert it into the Schrödinger equation:

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

After canceling the common factor $e^{i\alpha_n(t)}$, one obtains

$$i\hbar \left( i\dot\alpha_n\lvert n\rangle + \lvert \dot n\rangle \right) = E_n\lvert n\rangle.$$

Taking the inner product with $\langle n|$ gives

$$-\hbar\dot\alpha_n + i\hbar\langle n|\dot n\rangle = E_n.$$

Therefore

$$\dot\alpha_n = -\frac{E_n}{\hbar} + i\langle n|\dot n\rangle.$$

This is where the Berry term enters. It is not an extra force or a correction added by hand. It is the phase required by a changing eigenvector representative once transitions to other eigenspaces have been suppressed.

Adiabatic Basis View

A useful way to see the transition mechanism is to expand a general state in instantaneous eigenvectors:

$$\lvert\psi(t)\rangle = \sum_m c_m(t) \exp\left( -\frac{i}{\hbar} \int_0^t E_m(t')\,dt' \right) \lvert m(t)\rangle.$$

The changing basis produces couplings involving $\langle m|\dot k\rangle$. The off-diagonal terms with $m\ne k$ drive transitions between instantaneous eigenspaces. Adiabaticity means these transition amplitudes are small after their oscillatory phases and gap denominators are taken into account.

The diagonal terms $\langle n|\dot n\rangle$ do not move population to another eigenspace. They determine the geometric phase convention and, for a closed loop, the Berry holonomy.

What Follows: Ray or Vector?

The theorem says that the state remains in the corresponding eigenspace, not that it stays equal to the initial ket. Even in the one-dimensional case, the physical instantaneous state is a ray:

$$\lvert n(t)\rangle \sim e^{i\chi(t)}\lvert n(t)\rangle.$$

Changing the phase convention changes the integral

$$i\int_0^t\langle n|\dot n\rangle\,dt,$$

but it also changes the endpoint representative. The physical state and closed-loop phase factor are unchanged after the endpoint comparison is made.

For a degenerate but isolated eigenspace, the state follows a subspace rather than a single line. The geometric phase becomes a unitary matrix acting within that subspace; see Non-Abelian Berry Phase Preview.

Example: Spin Following a Magnetic Field

For

$$H(t) = -\frac{\Delta}{2} \hat{\mathbf n}(t)\cdot\boldsymbol\sigma, \qquad \Delta>0,$$

the instantaneous gap is $\Delta$. A schematic adiabatic condition is

$$\hbar\lvert\dot{\hat{\mathbf n}}\rvert \ll \Delta,$$

up to angular factors and the precise off-diagonal matrix element. If the direction $\hat{\mathbf n}(t)$ traces a closed loop, the state can follow the instantaneous spin eigenstate and acquire the solid-angle Berry phase worked out in Berry Phase for Spin-1/2.

If the field direction changes too quickly, the spin need not follow the instantaneous eigenstate. Then the simple Berry-phase formula is replaced by ordinary driven two-level dynamics, nonadiabatic transitions, or a different cyclic phase.

Example: Avoided Crossings

The Landau-Zener transition is the standard warning example. A two-level system swept through an avoided crossing may either follow an adiabatic branch or jump nonadiabatically to the other branch.

The minimum gap controls the difficulty. Slower sweeps and larger gaps suppress nonadiabatic transitions. Near a true crossing, the nondegenerate adiabatic picture fails because there is no isolated eigenline to follow.

Common Failure Modes

• The path crosses a degeneracy or the gap closes.
• The Hamiltonian changes too quickly near a small gap.
• The control path is not smooth enough for the theorem being used.
• Several nearby levels participate, so a two-level or one-level picture is incomplete.
• The evolution is open or effectively non-Hermitian, requiring a different adiabatic framework.
• The process is cyclic in controls but not adiabatic in state space.
• A gauge-dependent partial phase is mistaken for an observable closed-loop phase.

Common Mistakes

• Treating “slow” as an absolute word instead of comparing a rate with a gap scale.
• Saying the state “does not change” during adiabatic evolution. It changes by following an eigenspace and accumulating phase.
• Omitting the geometric phase when writing the adiabatic state.
• Applying the nondegenerate Berry formula through a degeneracy.
• Assuming the adiabatic diagnostic alone is a rigorous error bound in every infinite-dimensional problem.
• Forgetting that the theorem controls leakage between eigenspaces, while Berry phase describes transport within an eigenline or eigenspace.

Cross-Links

• Adiabatic Theorem
• Adiabatic Approximation as a Method
• Dynamical Phase versus Geometric Phase
• Berry Phase
• Berry Connection
• Berry Phase for Spin-1/2
• Non-Abelian Berry Phase Preview
• Time-Dependent Hamiltonians
• Time Ordering
• Landau-Zener Transition
• Projective Hilbert Space

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.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167-2170, 1983.
• 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.
• S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Springer, 2003.

Exercises

1. Derive the identity

$$\langle m|\dot n\rangle = \frac{\langle m|\dot H|n\rangle}{E_n-E_m}$$

for $m\ne n$, assuming nondegenerate instantaneous eigenstates.

Solution

Differentiate the eigenvalue equation:

$$\dot H\lvert n\rangle + H\lvert\dot n\rangle = \dot E_n\lvert n\rangle + E_n\lvert\dot n\rangle.$$

Take the inner product with $\langle m|$ for $m\ne n$. Since $\langle m|n\rangle=0$ and $\langle m|H=E_m\langle m|$,

$$\langle m|\dot H|n\rangle + E_m\langle m|\dot n\rangle = E_n\langle m|\dot n\rangle.$$

Thus

$$\langle m|\dot n\rangle = \frac{\langle m|\dot H|n\rangle}{E_n-E_m}.$$

2. Starting from $\lvert\psi(t)\rangle=e^{i\alpha(t)}\lvert n(t)\rangle$, derive

$$\dot\alpha = -\frac{E_n}{\hbar} + i\langle n|\dot n\rangle.$$

Solution

Insert the ansatz into the Schrödinger equation:

$$i\hbar \frac{d}{dt} \left( e^{i\alpha}\lvert n\rangle \right) = e^{i\alpha}E_n\lvert n\rangle.$$

Cancel $e^{i\alpha}$:

$$i\hbar \left( i\dot\alpha\lvert n\rangle + \lvert\dot n\rangle \right) = E_n\lvert n\rangle.$$

Taking $\langle n|$ and using $\langle n|n\rangle=1$ gives

$$-\hbar\dot\alpha + i\hbar\langle n|\dot n\rangle = E_n,$$

so

$$\dot\alpha = -\frac{E_n}{\hbar} + i\langle n|\dot n\rangle.$$

3. A spin-$1/2$ Hamiltonian has fixed gap $\Delta$ and its field direction rotates with characteristic angular speed $\omega$. What is the schematic adiabatic condition?

Solution

The rate of change of the direction is of order $\omega$, so the condition is

$$\hbar\omega \ll \Delta.$$

More precise statements include matrix elements and angular factors, but the gap-rate comparison is the essential scale estimate.

4. Why is the partial geometric phase on an open interval gauge dependent?

Solution

Under

$$\lvert n(t)\rangle \mapsto e^{i\chi(t)}\lvert n(t)\rangle,$$

the integrand changes as

$$i\langle n|\dot n\rangle \mapsto i\langle n|\dot n\rangle-\dot\chi.$$

Therefore

$$i\int_0^t\langle n|\dot n\rangle\,dt' \mapsto i\int_0^t\langle n|\dot n\rangle\,dt' - \chi(t)+\chi(0).$$

On an open interval this endpoint change matters unless an endpoint phase convention or interferometric comparison is specified. For a closed loop with a single-valued gauge, the change is an integer multiple of $2\pi$ in the phase.