Adiabatic Theorem - Proof of The Adiabatic Theorem

Proof of The Adiabatic Theorem

The first proof of this theorem was given by Max Born and Vladimir Fock, in Zeitschrift für Physik 51, 165 (1928). The concept of this theorem deals with the time-dependent Hamiltonian (which might be called a subject of Quantum dynamics) where the Hamiltonian changes with time.

For the case of a time-independent Hamiltonian or in a broad sense time-independent potential (subjects of Quantum statics) the Schrödinger equation:

can be simplified to the time-independent Schrödinger equation,

as the general solution of the Schrödinger equation then can be found by the method of Separation of variables to give the wavefunction of the form:

or, for nth eigenstate only :

This signifies that a particle which starts from the nth eigenstate remains in the nth eigenstate, simply picking up a phase factor .

In an adiabatic process the Hamiltonian is time-dependent i.e, the Hamiltonian changes with time (not to be confused with Perturbation theory, as here the change in the Hamiltonian is not small; it's huge, although it happens gradually). As the Hamiltonian changes with time, the eigenvalues and the eigenfunctions are time dependent.

But at any particular instant of time the states still form a Complete orthogonal system. i.e,

Notice that: The dependence on position is tacitly suppressed, as the time dependence part will be in more concern. will considered to be the state of the system at time t no-matter how it depends on its position.

The general solution of time dependent Schrödinger equation now can be expressed 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 hand side cancels out with the right hand side leaving

now taking the inner product with an arbitrary eigenfunction, the on the left gives which is 1 only for m = n otherwise vanishes. The remaining part gives

calculating the expression for from differentiating the modified time independent Schrödinger equation above it can have the form

This is also exact.
For the adiabatic approximation which says the time derivative of Hamiltonian i.e, is extremely small as time is largely taken, the last term will drop out and one has

that gives, after solving,

having defined the geometric phase as . Putting it in the expression for nth eigenstate one has

So, for an adiabatic process, a particle 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.