Adiabatic theorem

Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows: In simpler terms, a quantum mechanical system subjected to gradually changing external conditions adapts its functional form, but when subjected to rapidly varying conditions there is insufficient time for the functional form to adapt, so the spatial probability density remains unchanged.

At the 1911 Solvay conference, Einstein gave a lecture on the quantum hypothesis, which states that

, with a modified configuration: The degree to which a given change approximates an adiabatic process depends on both the energy separation between

we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged: The so-called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the spectrum of

The classical and quantum mechanics definition[8] is instead closer to the thermodynamical concept of a quasistatic process, which are processes that are almost always at equilibrium (i.e. that are slower than the internal energy exchange interactions time scales, namely a "normal" atmospheric heat wave is quasi-static, and a pressure wave is not).

A gradual change in external conditions allows the system to adapt, such that it retains its initial character.

[10] The classical nature of a pendulum precludes a full description of the effects of the adiabatic theorem.

Classically this is equivalent to increasing the stiffness of a spring; quantum-mechanically the effect is a narrowing of the potential energy curve in the system Hamiltonian.

For a rapidly increased spring constant, the system undergoes a diabatic process

For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field.

Assuming the magnetic-field dependence is linear, the Hamiltonian matrix for the system with the field applied can be written where

Figure 2 shows the dependence of the diabatic and adiabatic energies on the value of the magnetic field; note that for non-zero coupling the eigenvalues of the Hamiltonian cannot be degenerate, and thus we have an avoided crossing.

This coupled first-order differential equation is exact and expresses the time-evolution of the coefficients in terms of inner products

To do so, differentiate both sides of the time-independent Schrödinger equation with respect to time using the product rule to get

This differential equation describes the time-evolution of the coefficients, but now in terms of matrix elements of

And C is the path in the parameter space, This is the same as the statement of the theorem but in terms of the coefficients of the total wave function and its initial state.

Finally integrals of local geometric quantities can give topological invariants as in the case of the Gauss-Bonnet theorem.

[16] In fact if the path C is closed then the Berry phase persists to Gauge transformation and becomes a physical quantity.

Often a solid crystal is modeled as a set of independent valence electrons moving in a mean perfectly periodic potential generated by a rigid lattice of ions.

can be written where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the position operator It is instructive to examine the limiting cases, in which

Consider a system Hamiltonian undergoing continuous change from an initial value

can be obtained using The problem of determining the adiabaticity of a given process is equivalent to establishing the dependence of

To determine the validity of the adiabatic approximation for a given process, one can calculate the probability of finding the system in a state other than that in which it started.

is the root mean square deviation of the system Hamiltonian averaged over the interval of interest.

(the probability of finding the system in a state other than that in which is started approaches zero), thus the validity condition is given by which is a statement of the time-energy form of the Heisenberg uncertainty principle.

we have infinitely rapid, or diabatic passage: The functional form of the system remains unchanged: This is sometimes referred to as the sudden approximation.

The validity of the approximation for a given process can be characterized by the probability that the state of the system remains unchanged: In the limit

The validity of the approximation for a given process can be determined from the probability that the final state of the system is different from the initial state: In 1932 an analytic solution to the problem of calculating adiabatic transition probabilities was published separately by Lev Landau and Clarence Zener,[19] for the special case of a linearly changing perturbation in which the time-varying component does not couple the relevant states (hence the coupling in the diabatic Hamiltonian matrix is independent of time).

For a transition involving a nonlinear change in perturbation variable or time-dependent coupling between the diabatic states, the equations of motion for the system dynamics cannot be solved analytically.

The diabatic transition probability can still be obtained using one of the wide varieties of numerical solution algorithms for ordinary differential equations.

Figure 1. Change in the probability density, , of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.
Figure 2. An avoided energy-level crossing in a two-level system subjected to an external magnetic field. Note the energies of the diabatic states, and and the eigenvalues of the Hamiltonian, giving the energies of the eigenstates and (the adiabatic states). (Actually, and should be switched in this picture.)