6. Particles in a Magnetic Field The purpose of this chapter is to understand how quantum particles react to magnetic fields. Before we get to describe quantum effects, we first need to highlight a few of the more subtle aspects that arise when discussing classical physics in the presence of a magnetic field. 6.1 Gauge Fields Recall from our lectures on Electromagnetism that the electric field E(x, t) and mag- netic field B(x, t) can be written in terms a scalar potential ø(x, t) and a vector potential A(x,t), E = - V? - JA and B = V x A (6.1) at Both o and A are referred to as gauge fields. When we first learn electromagnetism, they are introduced merely as handy tricks to help solve the Maxwell equations. However, as we proceed through theoretical physics, we learn that they play a more fundamental role. In particular, they are necessary if we want to discuss a Lagrangian or Hamiltonian approach to electromagnetism. We will soon see that these gauge fields are quite indispensable in quantum mechanics. The Lagrangian for a particle of charge q and mass m moving in a background electromagnetic fields is L = mx2 + qx · A -q¢ (6.2) The classical equation of motion arising from this Lagrangian is mx = q (E + * x B) This is the Lorentz force law. Before we proceed I should warn you of a minus sign issue. We will work with a general charge q. However, many textbooks work with the charge of the electron, written as q = - e. If this minus sign leans to confusion, you should blame Benjamin Franklin. - 164 -
An Example: Motion in a Constant Magnetic Field We'll take a constant magnetic field, pointing in the z-direction: B = (0, 0, B). We'll take E = 0. The particle is free in the z-direction, with the equation of motion mž = 0. The more interesting dynamics takes place in the (x, y)-plane where the equations of motion are mï = qBy and m?= - qBi (6.3) which has general solution is x(t) = X + Rsin(WB (t - to)) and y(t) = Y + R cos(WB (t - to)) We see that the particle moves in a circle which, for B > 0 and q > 0, is in a clockwise direction. The cyclotron frequency is defined by WB = q.B m (6.4) The centre of the circle (X, Y), the radius of the circle R and the phase t0 are all arbitrary. These are the four integration constants expected in the solution of two, second order differential equations. B Figure 78: 6.1.1 The Hamiltonian The canonical momentum in the presence of gauge fields is OL p = 2x = mx + qA (6.5) This clearly is not the same as what we naively call momentum, namely m x. The Hamiltonian is given by H = x . p - L = 2m 1 -(p - qA)2 + qø