An adiabatic system is a system whose Hamiltonian changes slowly in
time. Despite the time dependence of the Hamiltonian, the wave
function can still be written in terms of the energy eigenfunctions
In particular, in the adiabatic approximation, the wave function of
a system can be written as, {D.34}:
Note that if the Hamiltonian does not depend on time, the above
expression simplifies to the usual solution of the Schrödinger equation as
given in chapter 7.1.2. In particular, in that case the
geometric phase is zero and the dynamic phase is the usual
Even if the Hamiltonian depends on time, the geometric phase is still
zero as long as the Hamiltonian is real. The reason is that real
Hamiltonians have real eigenfunctions; then
If the geometric phase is nonzero, you may be able to play games with
it. Suppose first that Hamiltonian changes with time because some
single parameter
But now suppose that not one, but a set of parameters
You might assume that it is irrelevant since the phase of the wave function is not observable anyway. But if a beam of particles is sent along two different paths, the phase difference between the paths will produce interference effects when the beams merge again.
Systems that do not return to the same state when they are taken
around a closed loop are not just restricted to quantum mechanics. A
classical example is the Foucault pendulum, whose plane of oscillation
picks up a daily angular deviation when the motion of the earth
carries it around a circle. Such systems are called “nonholonomic” or anholonomic.