Short description: Concept in quantum mechanics

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The adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. In simpler terms, a quantum-mechanical system subjected to gradually changing external conditions adapts its functional form, whereas under rapidly varying conditions there is insufficient time for the state to adapt, so the spatial probability density remains unchanged.^[1]

Introduction

The theorem in its original form was given by Max Born and Vladimir Fock in 1928.^[1] It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.

The theorem is closely related to the notion of adiabatic invariant in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word adiabatic. In quantum mechanics, adiabatic generally means slow variation compared with the system’s internal timescales.^[2]

Adiabatic pendulum

At the 1911 Solvay conference, Albert Einstein discussed the quantum hypothesis that for atomic oscillators $E = n h ν$ . After Einstein's lecture, Hendrik Lorentz pointed out that if a classical pendulum is shortened gradually, its energy appears to change smoothly. Einstein replied that although both the energy $E$ and the frequency $ν$ vary, their ratio $\frac{E}{ν}$ remains conserved, preserving the quantum hypothesis.^[3]

Before the conference Einstein had read a paper by Paul Ehrenfest on the adiabatic hypothesis, and later referred to it in correspondence with Michele Besso.^[4]^[5]^[6]

Diabatic vs. adiabatic processes

Rapidly changing conditions prevent the system from adapting its configuration during the process, so the spatial probability density remains unchanged. The system then generally ends in a linear superposition of eigenstates of the final Hamiltonian. By contrast, gradually changing conditions allow the probability density to change continuously, and if the system begins in an eigenstate of the initial Hamiltonian it ends in the corresponding eigenstate of the final Hamiltonian.^[7]

At an initial time $t_{0}$ a quantum-mechanical system has Hamiltonian $\hat{H} (t_{0})$ and is in an eigenstate $ψ (x, t_{0})$ . A continuous change in conditions modifies the Hamiltonian to $\hat{H} (t_{1})$ at time $t_{1}$ . The theorem states that the behavior depends critically on the interval $τ = t_{1} - t_{0}$ over which the change occurs.

For a truly adiabatic process, $τ \to \infty$ , and the final state becomes an eigenstate of the final Hamiltonian with a modified spatial form: $| ψ (x, t_{1}) |^{2} \neq | ψ (x, t_{0}) |^{2} .$

Conversely, in the limit $τ \to 0$ , corresponding to infinitely rapid or diabatic passage, the configuration remains unchanged: $| ψ (x, t_{1}) |^{2} = | ψ (x, t_{0}) |^{2} .$

The approximation depends on both the energy gap separating the relevant state from nearby states and the ratio of the process duration $τ$ to the internal timescale $τ_{int} = 2 π ℏ / E_{0}$ .

The original theorem assumed a discrete and nondegenerate spectrum, so that one can identify which final eigenstate corresponds to the initial one. Later work generalized the theorem to cases without a strict gap condition.^[8]

Comparison with the adiabatic concept in thermodynamics

In thermodynamics, an adiabatic process is one in which there is no exchange of heat between system and environment. In practice, such processes are often relatively fast compared with the timescale of heat transfer.

In classical and quantum mechanics, however, adiabatic means something closer to a quasistatic process: a process slow enough that the system remains near equilibrium or stays in the corresponding instantaneous eigenstate.^[2]

Thus, in quantum mechanics the adiabatic theorem states that quantum jumps are suppressed when the Hamiltonian changes sufficiently slowly, and the system tends to preserve its state label and quantum numbers.^[9]

Example systems

Simple pendulum

A useful classical analogy is a pendulum whose support is moved slowly. If the change is gradual enough, the mode of oscillation relative to the support remains essentially unchanged, showing how a slow external variation allows the system to adapt continuously.^[10]

Quantum harmonic oscillator

A more specifically quantum example is the quantum harmonic oscillator with increasing spring constant $k$ . If $k$ increases adiabatically, the system remains in the corresponding instantaneous eigenstate of the current Hamiltonian. For a state initially in the ground state, the wavefunction narrows as the potential steepens, but the quantum number remains unchanged.

If $k$ increases rapidly, the process becomes diabatic. The system then has no time to adapt to the changing potential and the final state becomes a superposition of many eigenstates of the new Hamiltonian, whose combination reproduces the original probability density.

Avoided curve crossing

A particularly important example is a two-level atom in an external magnetic field. The system may be described in terms of diabatic states $| 1 ⟩$ and $| 2 ⟩$ with wavefunction $| Ψ ⟩ = c_{1} (t) | 1 ⟩ + c_{2} (t) | 2 ⟩ .$

If the magnetic-field dependence is linear, the Hamiltonian matrix may be written $𝐇 = (\begin{matrix} μ B (t) - ℏ ω_{0} / 2 & a \\ a^{*} & ℏ ω_{0} / 2 - μ B (t) \end{matrix})$

with eigenvalues $\begin{aligned} ε_{1} (t) & = - \frac{1}{2} \sqrt{4 a^{2} + (ℏ ω_{0} - 2 μ B (t))^{2}} \\ ε_{2} (t) & = + \frac{1}{2} \sqrt{4 a^{2} + (ℏ ω_{0} - 2 μ B (t))^{2}} . \end{aligned}$

Because of the off-diagonal coupling, the eigenvalues do not cross but exhibit an avoided crossing. Under slow variation of the magnetic field, the system follows an adiabatic eigenstate. Under rapid variation it instead follows a diabatic path, giving a finite probability of transition between the two eigenstates.^[11]

Mathematical statement

Under a slowly changing Hamiltonian $H (t)$ with instantaneous eigenstates $| n (t) ⟩$ and corresponding energies $E_{n} (t)$ , a quantum system evolves from $| ψ (0) ⟩ = \sum_{n} c_{n} (0) | n (0) ⟩$ to $| ψ (t) ⟩ = \sum_{n} c_{n} (t) | n (t) ⟩,$ where the coefficients change only by phase factors, $c_{n} (t) = c_{n} (0) e^{i θ_{n} (t)} e^{i γ_{n} (t)} .$

The dynamical phase is $θ_{m} (t) = - \frac{1}{ℏ} \int_{0}^{t} E_{m} (t^{'}) d t^{'}$

and the geometric phase is $γ_{m} (t) = i \int_{0}^{t} ⟨ m (t^{'}) | \dot{m} (t^{'}) ⟩ d t^{'} .$

Hence $| c_{n} (t) |^{2} = | c_{n} (0) |^{2}$ . In particular, if the system starts in an eigenstate of $H (0)$ , it remains in the corresponding eigenstate of $H (t)$ throughout the evolution, apart from phase changes only.

Proofs

A standard derivation expands the full state in the instantaneous eigenbasis of the time-dependent Hamiltonian and inserts that expansion into the time-dependent Schrödinger equation. This yields an exact set of coupled first-order differential equations for the coefficients. The off-diagonal couplings are proportional to matrix elements of $\dot{H} (t)$ divided by energy differences between eigenstates. If the Hamiltonian changes slowly enough and a finite gap is maintained, these off-diagonal terms can be neglected, leading directly to the adiabatic approximation and to the phase factors above.^[12]

A more detailed formulation expresses the slowly varying Hamiltonian as $\hat{H} (t / T)$ and studies the limit $T \to \infty$ . In this limit, oscillatory off-diagonal terms are suppressed and the exact solution approaches the adiabatic one with an error of order $O (T^{- 1})$ . For cyclic adiabatic evolution, the geometric phase becomes a gauge-invariant physical observable, known as the Berry phase.^[13]^[14]

A parameter-space formulation writes the Hamiltonian as $H (\vec{R} (t))$ , where $\vec{R}$ varies slowly in time. In this approach the Berry phase can be written as a line integral along a path in parameter space, $γ_{m} (t) = i \int_{C} ⟨ ψ_{m} (\vec{R}) | \partial_{\vec{R}} | ψ_{m} (\vec{R}) ⟩ d \vec{R},$ which highlights its geometric character and its relation to topological ideas.^[15]^[16]

Example applications

The adiabatic theorem underlies many approximation schemes in quantum physics, especially those involving a separation of fast and slow degrees of freedom. A prominent example is the Born–Oppenheimer approximation, where fast electronic motion and slow ionic motion in crystals or molecules are treated separately.^[17]

This helps explain a variety of phenomena in:

thermodynamics: temperature dependence of specific heat, thermal expansion, and melting
transport phenomena: temperature dependence of electric resistivity in conductors, temperature dependence of electric conductivity in insulators, and properties of low-temperature superconductivity
optics: infrared absorption in ionic crystals, Brillouin scattering, and Raman scattering

Deriving conditions for diabatic vs adiabatic passage

One way to analyze the validity of the adiabatic approximation is through the time-evolution operator $\hat{U} (t, t_{0})$ , defined by $i ℏ \frac{\partial}{\partial t} \hat{U} (t, t_{0}) = \hat{H} (t) \hat{U} (t, t_{0}),$ with $\hat{U} (t_{0}, t_{0}) = 1 .$

The system evolves as $| ψ (t) ⟩ = \hat{U} (t, t_{0}) | ψ (t_{0}) ⟩ .$

To test whether a process is sudden or adiabatic, one can define the probability of leaving the initial state. In the perturbative limit this leads to $ζ = \frac{τ^{2} Δ {\bar{H}}^{2}}{ℏ^{2}},$ where $\bar{H}$ is the Hamiltonian averaged over the interval and $Δ \bar{H}$ is its root-mean-square deviation.

The sudden approximation is valid when $τ ≪ \frac{ℏ}{Δ \bar{H}},$ which is a form of the time-energy uncertainty relation.^[18]

Diabatic passage

In the limit $τ \to 0$ , $\lim_{τ \to 0} \hat{U} (t_{1}, t_{0}) = 1 .$

The functional form of the state remains unchanged: $| ⟨ x | ψ (t_{1}) ⟩ |^{2} = {| ⟨ x | ψ (t_{0}) ⟩ |}^{2} .$

This is the sudden approximation, and the probability that the system remains unchanged is $P_{D} = 1 - ζ .$

Adiabatic passage

In the limit $τ \to \infty$ , the system has time to adapt continuously to the changing Hamiltonian: $| ⟨ x | ψ (t_{1}) ⟩ |^{2} \neq | ⟨ x | ψ (t_{0}) ⟩ |^{2} .$

If the system begins in an eigenstate of $\hat{H} (t_{0})$ , it evolves into the corresponding eigenstate of $\hat{H} (t_{1})$ . The probability associated with adiabatic passage is $P_{A} = ζ .$

Calculating adiabatic passage probabilities

The Landau–Zener formula

For a two-level system with a linearly changing perturbation, the probability of a diabatic transition can be estimated using the Landau–Zener formula. The key quantity is the Landau–Zener velocity $v_{LZ} = \frac{\frac{\partial}{\partial t} | E_{2} - E_{1} |}{\frac{\partial}{\partial q} | E_{2} - E_{1} |} \approx \frac{d q}{d t},$ where $q$ is the perturbation parameter.

The diabatic transition probability is then $\begin{aligned} P_{D} & = e^{- 2 π Γ} \\ Γ & = \frac{a^{2} / ℏ}{| \frac{\partial}{\partial t} (E_{2} - E_{1}) |} = \frac{a^{2} / ℏ}{| \frac{d q}{d t} \frac{\partial}{\partial q} (E_{2} - E_{1}) |} \\ = \frac{a^{2}}{ℏ | α |} . \end{aligned}$

A large sweep rate gives a large diabatic transition probability, whereas a slow sweep favors adiabatic following.^[19]

The numerical approach

For nonlinear changes in the perturbation variable or time-dependent couplings, the system generally cannot be solved analytically and one must use numerical methods for ordinary differential equations. The amplitudes satisfy $i ℏ {\dot{\underline{c}}}^{A} (t) = 𝐇_{A} (t) {\underline{c}}^{A} (t),$ where ${\underline{c}}^{A} (t)$ is the vector of adiabatic-state amplitudes and $𝐇_{A} (t)$ is the adiabatic Hamiltonian.^[11]