Adiabatic Theorem Reminder

Adiabatic evolution suppresses transitions out of an isolated eigenspace while its parameters change. The state still moves through Hilbert space and accumulates dynamical and geometric phases. The adiabatic theorem states precise hypotheses; adiabatic approximation develops runtime estimates. Here the focus is the separation between eigenstate following, leakage, and phase; Berry phase derives the geometric term from evolution.

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.

Adiabatic Eigenstate Following

For a finite-dimensional smooth family $H(s)$, $0\le s\le1$, suppose the chosen eigenvalue has a positive minimum gap along the entire fixed path. Run the schedule $s=t/T$. Under the usual smoothness assumptions, the state approaches the instantaneous eigenline as $T$ grows:

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

when the initial state is $\lvert n(0)\rangle$, with $\sup_{0\le t\le T}\|\delta\psi(t)\|\to0$ as $T\to\infty$. This error is not automatically equal to the local diagnostic below. The leading 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 loop it gives the Berry phase when the eigenvector gauge is periodic; otherwise the endpoint phase $\arg\langle n(0)|n(T)\rangle$ must be added.

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

The phase derivation projects the Schrödinger equation onto the followed eigenstate and obtains

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

Its orthogonal projection supplies the transition terms instead. The diagonal phase equation and suppression of those transitions are different parts of the argument: solving the first does not justify neglecting the second.

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

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

For this two-level model, the precise instantaneous mixing ratio is $\hbar|\dot{\hat{\mathbf n}}|/(2\Delta)$. At fixed polar angle with azimuthal speed $\omega$, it is $\hbar|\omega\sin\theta|/(2\Delta)$; at a pole the Hamiltonian does not change at all. 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 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.

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. Rapid phase convention, stationary eigenspace. Let $H$ be constant with normalized eigenstate $|n_0\rangle$ and energy $E_n$. Choose the representative $|n(t)\rangle=e^{i\kappa t}|n_0\rangle$ for an arbitrarily large real $\kappa$. Compute $\|\dot n\|$, the transition matrix elements to orthogonal eigenstates, and the phase $\alpha_n(t)$. Does this representative make the physical evolution nonadiabatic?

Solution

Although $\|\dot n\|=|\kappa|$, the entire derivative is parallel to $|n\rangle$. Therefore $\langle m|\dot n\rangle=0$ for $m\ne n$, and the projector is constant. The diagonal connection is $i\langle n|\dot n\rangle=-\kappa$, so

$$\alpha_n(t)=-\frac{E_nt}{\hbar}-\kappa t, \qquad e^{i\alpha_n(t)}|n(t)\rangle=e^{-iE_nt/\hbar}|n_0\rangle.$$

There is no leakage, regardless of $\kappa$. The physical motion is exactly stationary in ray space. The derivative’s norm alone is gauge dependent; the off-diagonal derivative or the moving spectral projector diagnoses changes of eigenspace. On a closed interval the endpoint phase $\kappa T$ also cancels the connection integral $-\kappa T$, giving zero Berry phase modulo $2\pi$.

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.

References

• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45–57, 1984.
• 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.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167–2170, 1983.
• S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Springer, 2003.