Adiabatic theorem

The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock, 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.

Adiabatic pendulum

At the 1911 Solvay conference, Einstein gave a lecture on the quantum hypothesis, which states that for atomic oscillators. After Einstein's lecture, Hendrik Lorentz commented that, classically, if a simple pendulum is shortened by holding the wire between two fingers and sliding down, it seems that its energy will change smoothly as the pendulum is shortened. This seems to show that the quantum hypothesis is invalid for macroscopic systems, and if macroscopic systems do not follow the quantum hypothesis, then as the macroscopic system becomes microscopic, it seems the quantum hypothesis would be invalidated. Einstein replied that although both the energy and the frequency would change, their ratio would still be conserved, thus saving the quantum hypothesis.
Before the conference, Einstein had just read a paper by Paul Ehrenfest on the adiabatic hypothesis. We know that he had read it because he mentioned it in a letter to Michele Besso written before the conference.

Diabatic vs. adiabatic processes

At some initial time a quantum-mechanical system has an energy given by the Hamiltonian ; the system is in an eigenstate of labelled. Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian at some later time. The system will evolve according to the time-dependent Schrödinger equation, to reach a final state. The adiabatic theorem states that the modification to the system depends critically on the time during which the modification takes place.
For a truly adiabatic process we require ; in this case the final state will be an eigenstate of the final Hamiltonian, with a modified configuration:
The degree to which a given change approximates an adiabatic process depends on both the energy separation between and adjacent states, and the ratio of the interval to the characteristic timescale of the evolution of for a time-independent Hamiltonian,, where is the energy of.
Conversely, in the limit 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 is discrete and nondegenerate, such that there is no ambiguity in the ordering of the states. In 1999 J. E. Avron and A. Elgart reformulated the adiabatic theorem to adapt it to situations without a gap.

Comparison with the adiabatic concept in thermodynamics

The term "adiabatic" is traditionally used in thermodynamics to describe processes without the exchange of heat between system and environment, more precisely these processes are usually faster than the timescale of heat exchange. Adiabatic in the context of thermodynamics is often used as a synonym for fast process.
The classical and quantum mechanics definition is instead closer to the thermodynamical concept of a quasistatic process, which are processes that are almost always at equilibrium. Adiabatic in the context of mechanics is often used as a synonym for slow process.
In the quantum world adiabatic means for example that the time scale of electrons and photon interactions is much faster or almost instantaneous with respect to the average time scale of electrons and photon propagation. Therefore, we can model the interactions as a piece of continuous propagation of electrons and photons plus a quantum jump between states.
The adiabatic theorem in this heuristic context tells essentially that quantum jumps are preferably avoided, and the system tries to conserve the state and the quantum numbers.
The quantum mechanical concept of adiabatic is related to adiabatic invariant, it is often used in the old quantum theory and has no direct relation with heat exchange.

Example systems

Simple pendulum

As an example, consider a pendulum oscillating in a vertical plane. If the support is moved, the mode of oscillation of the pendulum will change. If the support is moved sufficiently slowly, the motion of the pendulum relative to the support will remain unchanged. A gradual change in external conditions allows the system to adapt, such that it retains its initial character. The detailed classical example is available in the Adiabatic invariant page and here.

Quantum harmonic oscillator

The classical nature of a pendulum precludes a full description of the effects of the adiabatic theorem. As a further example consider a quantum harmonic oscillator as the spring constant is increased. 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.
If is increased adiabatically then the system at time will be in an instantaneous eigenstate of the current Hamiltonian, corresponding to the initial eigenstate of. For the special case of a system like the quantum harmonic oscillator described by a single quantum number, this means the quantum number will remain unchanged. Figure 1 shows how a harmonic oscillator, initially in its ground state,, remains in the ground state as the potential energy curve is compressed; the functional form of the state adapting to the slowly varying conditions.
For a rapidly increased spring constant, the system undergoes a diabatic process in which the system has no time to adapt its functional form to the changing conditions. While the final state must look identical to the initial state for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian,, that resembles the initial state. The final state is composed of a linear superposition of many different eigenstates of which sum to reproduce the form of the initial state.

Avoided curve crossing

For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field. The states, labelled and using bra–ket notation, can be thought of as atomic angular-momentum states, each with a particular geometry. For reasons that will become clear these states will henceforth be referred to as the diabatic states. The system wavefunction can be represented as a linear combination of the diabatic states:
With the field absent, the energetic separation of the diabatic states is equal to ; the energy of state increases with increasing magnetic field, while the energy of state decreases with increasing magnetic field. Assuming the magnetic-field dependence is linear, the Hamiltonian matrix for the system with the field applied can be written
where is the magnetic moment of the atom, assumed to be the same for the two diabatic states, and is some time-independent coupling between the two states. The diagonal elements are the energies of the diabatic states, however, as is not a diagonal matrix, it is clear that these states are not eigenstates of due to the off-diagonal coupling constant.
The eigenvectors of the matrix are the eigenstates of the system, which we will label and, with corresponding eigenvalues
It is important to realise that the eigenvalues and are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies and correspond to the expectation values for the energy of the system in the diabatic states and.
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. If an atom is initially in state in zero magnetic field, an adiabatic increase in magnetic field will ensure the system remains in an eigenstate of the Hamiltonian throughout the process. A diabatic increase in magnetic field will ensure the system follows the diabatic path, such that the system undergoes a transition to state. For finite magnetic field slew rates there will be a finite probability of finding the system in either of the two eigenstates. See [|below] for approaches to calculating these probabilities.
These results are extremely important in atomic and molecular physics for control of the energy-state distribution in a population of atoms or molecules.

Mathematical statement

Under a slowly changing Hamiltonian with instantaneous eigenstates and corresponding energies, a quantum system evolves from the initial stateto the final statewhere the coefficients undergo the change of phase
with the dynamical phase
and geometric phase
In particular,, so if the system begins in an eigenstate of, it remains in an eigenstate of during the evolution with a change of phase only.

Proofs

Adiabatic approximation

Proof with the details of the adiabatic approximation
We are going to formulate the statement of the theorem as follows:
And now we are going to prove the theorem.
Consider the time-dependent Schrödinger equation
with Hamiltonian
We would like to know the relation between an initial state and its final state at in the adiabatic limit
First redefine time as :
At every point in time can be diagonalized with eigenvalues and eigenvectors. Since the eigenvectors form a complete basis at any time we can expand as:
where
The phase is called the dynamic phase factor. By substitution into the Schrödinger equation, another equation for the variation of the coefficients can be obtained:
The term gives, and so the third term of left side cancels out with the right side, leaving
Now taking the inner product with an arbitrary eigenfunction, the on the left gives, which is 1 only for m = n and otherwise vanishes. The remaining part gives
For the will oscillate faster and faster and intuitively will eventually suppress nearly all terms on the right side. The only exceptions are when has a critical point, i.e.. This is trivially true for. Since the adiabatic theorem assumes a gap between the eigenenergies at any time this cannot hold for. Therefore, only the term will remain in the limit.
In order to show this more rigorously we first need to remove the term.
This can be done by defining
We obtain:
This equation can be integrated:
or written in vector notation
Here is a matrix and
is basically a Fourier transform.
It follows from the Riemann-Lebesgue lemma that as. As last step take the norm on both sides of the above equation:
and apply Grönwall's inequality to obtain
Since it follows for. This concludes the proof of the adiabatic theorem.
In the adiabatic limit the eigenstates of the Hamiltonian evolve independently of each other. If the system is prepared in an eigenstate its time evolution is given by:
So, for an adiabatic process, a system starting from nth eigenstate also remains in that nth eigenstate like it does for the time-independent processes, only picking up a couple of phase factors. The new phase factor can be canceled out by an appropriate choice of gauge for the eigenfunctions. However, if the adiabatic evolution is cyclic, then becomes a gauge-invariant physical quantity, known as the Berry phase.

Generic proof in parameter space

Let's start from a parametric Hamiltonian, where the parameters are slowly varying in time, the definition of slow here is defined essentially by the distance in energy by the eigenstates.
This way we clearly also identify that while slowly varying the eigenstates remains clearly separated in energy. Given they do not intersect the states are ordered and in this sense this is also one of the meanings of the name topological order.
We do have the instantaneous Schrödinger equation:
And instantaneous eigenstates:
The generic solution:
plugging in the full Schrödinger equation and multiplying by a generic eigenvector:
And if we introduce the adiabatic approximation:
for each
We have
and
where
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.
Now this is slightly more general than the other proofs given we consider a generic set of parameters, and we see that the Berry phase acts as a local geometric quantity in the parameter space.
Finally integrals of local geometric quantities can give topological invariants as in the case of the Gauss-Bonnet theorem.
In fact if the path C is closed then the Berry phase persists to gauge transformation and becomes a physical quantity.