Physics 221A: Berry's Phase Hints

Meaning of the geometric phase

A quantum state is also a vector, and if it depends on parameters of the Hamiltonian, it can be moved around in parameter space.  We will also assume that the parameters change adiabatically: the time dependence of the parameters contains only frequencies much lower than the frequencies corresponding to energy differences between the states of the Hamiltonian.  This means if the system starts in an eigenstate, it will not make a transition out of it.  But the system's state vector can acquire a geometric phase, and this means it can exhibit interference with another system that didn't undergo the change.  The Aharonov-Bohm effect is an example.

See also this nice introduction to Berry's phase with many examples. And this post about how to demonstrate Berry's phase with coffee cups.

Derivation of the geometric phase

H[R] |m> = E_m|m>

where the states |m> and their eigenvalues are also functions of the parameters R, which change with time.  Now let's assume that the system is in a particular state |m> at time t=0, and that its state at some later time has the form:

where the second and third factors represent the "dynamic" and "geometric" phases, respectively.  Plug this state into the Schrödinger equation:

so, after cancelling the common exponential factors,

Note that the eigenstates |m> would be independent of time if the parameters of the Hamiltonian did not change; their only time dependence is through the change of the parameters.  Next, subtract the E_m terms from both sides and then take the scalar product with <m|:

Therefore, the phase shift g due to a change in the parameters is

or equivalently, since the eigenstates |m> depend on time only through the parameters R,

which proves equation (3.48) in the lecture notes.

Calculations with a particular Hamiltonian

The example in the lecture notes and this problem set is a spin-1/2 particle in a magnetic field, whose Hamiltonian is

H = -S·B = -(1/2)h (B_xs_x+ B_ys_y+ B_zs_z) or H = Xs_x+ Ys_y+ Zs_z,

where s_x, s_y, s_z are the Pauli matrices.  The two eigenstates of the Hamiltonian represent the particle's spin aligned parallel or antiparallel to the magnetic field.  If the spin starts out aligned with the field, and the field is slowly changed, the direction of the spin will follow it: it remains in an eigenstate.

The first step is to find the eigenstates of the Hamiltonian and define them so they are differentiable with respect to the parameters over as much of the parameter space as possible.  For example, if f ranges from 0 to 2p, the eigenstates of the two-parameter Hamiltonian shown in the lecture notes (3.51) are not continuous at f=0, but the eigenstates (3.52) are continuous everywhere in the two-parameter space except at the origin.  Similarly, in the three-dimensional problem, you can define eigenstates that are continuous and differentiable everywhere in the parameter space except the origin and the negative z-axis (or, alternately, the positive z-axis).

Next, find the gradient of these eigenstates with respect to the parameters, and find the vector <m| |m>.  You need to find the line integral of this vector around any arbitrary path (some students asked what path to use; you're supposed to find a result that is true for any path).  Use Stokes' theorem; the curl of the vector field should be very simple, and should make the connection with the "flux through the image loop" obvious.  (However, the strength of the monopole should be +/- 1/2, depending on which eigenstate the system is in, not 1/4p.)

Finally, if you had defined the eigenstates another way, e.g. by excluding the positive z-axis instead of the negative z-axis, the final state of the system should not be affected.  Show that if you do this, you must choose a different surface spanning the loop, and g would change by 2p, which of course has no physical effect.