Classical and Quantum Mechanical Motion in Magnetic Fields

Joel Franklin, Katherine Newton

We study the motion of a particle in a particular magnetic field configuration both classically and quantum mechanically. For flux-free radially symmetric magnetic fields defined on circular regions, we establish that particle escape speeds depend, classically, on a gauge-fixed magnetic vector potential, and we demonstrate some trajectories associated with this special type of magnetic field. Then we show that some of the geometric features of the classical trajectory (perpendicular exit from the field region, trapped and escape behavior) are reproduced quantum mechanically, using a numerical method that extends the norm-preserving Crank-Nicolson method to problems involving magnetic fields. While there are similarities between the classical trajectory and the position expectation value of the quantum-mechanical solution, there are also differences, and we demonstrate some of these.

F. Suekane

Earle H. Kennard

Min Zhao

In Newtonian mechanics, the motion of the charge in the magnetic field is the equidistance helical line. It is composed of the uniform motion in magnetic field’s direction and the circular motion in vertical direction. In quantum mechanics, its motion is the two-dimensional resonance. Its energy level is separate. Its wave function is probability. With this theory, magnetic focusing is explained to that a large number of charge moving in the magnetic field is not 100% at a point. They will be dispersed according to probability.

Daniel Thies

Daniel F. Styer

How shall we approach the principles of quantum mechanics? One way is simply to write them down. In fact I have already done that (in the first paragraph of the Preface), but to do so I had to use words and concepts that you don't yet understand. To develop the necessary understanding I will use a particular physical system as a vehicle to propel our exploration of quantum mechanics. Which system? An obvious choice is the motion of a tossed ball. Unfortunately this system, while simple and familiar in classical mechanics, is a complicated one in quantum mechanics. We will eventually get to the quantum mechanics of a tossed ball (in chapter 14, “Quantum mechanics of a bouncing ball”, page 103), but as the vehicle for developing quantum mechanics I will instead use a system that is simple in quantum mechanics but that is, unfortunately, less familiar in daily life. That system is the magnetic needle in a magnetic field. This chapter describes the classical motion of a magnetic needle so that we will be able to see how its classical and quantal behaviors differ. Magnetic needle in a magnetic field A magnetic needle — like the one found in any woodsman's compass — has a “north pole” and a “south pole”. I will symbolize the magnetic needle by an arrow pointing from its south pole to its north pole.