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Wikipedia / Adiabatic theorem

The adiabatic theorem is a concept in quantum mechanics. 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.

Adiabatic pendulum

At the 1911 Solvay conference for atomic oscillators. After Einstein's lecture, Hendrik Lorentz 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

. The system will evolve according to the time-dependent Schrödinger equation during which the modification takes place.

, with a modified configuration:

.

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 is discrete and nondegenerate corresponds). 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 (see adiabatic process), more precisely these processes are usually faster than the timescale of heat exchange. For example, a pressure wave is adiabatic with respect to a heat wave, which is not adiabatic. 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 (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). 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 (i.e. states at equilibrium) plus a quantum jump between states (i.e. instantaneous).

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.

of the current. 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.

, that resembles the initial state. The final state is composed of a linear superposition 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. 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:

decreases with increasing magnetic field (a high-field-seeking state). Assuming the magnetic-field dependence is linear, the Hamiltonian matrix for the system with the field applied can be written

is the magnetic moment is some time-independent coupling is not a diagonal matrix due to the off-diagonal coupling constant.

correspond to the expectation values.

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 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

with the dynamical phase

and geometric phase

during the evolution with a change of phase only.

Proofs

Example applications

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. With the Adiabatic theorem we can also include instead the motion of the valence electrons across the crystal and the thermal motion of the ions as in the Born–Oppenheimer approximation.

This does explain many phenomena in the scope of:

Deriving conditions for diabatic vs adiabatic passage

We will now pursue a more rigorous analysis. Making use of bra–ket notation, the state vector can be written

where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the position operator

is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change).

. The evolution of the system can be described in the Schrödinger picture by the time-evolution operator, defined by the integral equation

which is equivalent to the Schrödinger equation.

along with the initial condition. Given knowledge of the system wave function can be obtained using

The problem of determining the adiabaticity.

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. Using bra–ket notation, we have:

In the perturbative limit, recognizing that

, we have:

After expanding the products and making the appropriate cancellations, we are left with:

giving

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.

Diabatic passage

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:

Adiabatic passage

we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,

If the system is initially in an eigenstate it will have passed into the corresponding.

This is referred to as the adiabatic approximation. 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:

Calculating adiabatic passage probabilities

The Landau–Zener formula

In 1932 an analytic solution to the problem of calculating adiabatic transition probabilities was published separately by Lev Landau and Clarence Zener, 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).

results in a large diabatic transition probability and vice versa.

, of a diabatic transition is given by

The numerical approach

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.

The equations to be solved can be obtained from the time-dependent Schrödinger equation:

is a vector is the time-dependent adiabatic Hamiltonian, and the overdot represents a time derivative.

.

See also

References