senior research project, 1997-98

Ordinary matter is composed of atoms, which are essentially electrons hovering in a cloud about the atomic nucleus. The nucleus is itself a bundle of smaller particles known as protons and neutrons. These basic particles behave in ordinary matter very much like indivisible dots, and can be described in terms of a few simple properties, including electric charge, mass, and spin. Spin is an intrinsic property of all elementary particles that describes how many distinct orientations the particle possesses. Some particles are so perfectly spherical that all orientations are equivalent; they are said to carry zero spin. These particles are relatively rare in nature. A particle which has two orientations (call them up and down) is called spin-1/2. Electrons, protons, and neutrons are all spin-1/2 particles.

The limited number of possible orientations for elementary particles, referred to as ``space quantisation'', is a unique feature of quantum physics, where the normal freedom that objects have to be oriented in any direction is restricted to a discrete set of (2S+1) orientations where S is spin. The first bona fide experimental evidence for intrinsic spin was published ``Experimental Proof of Space Quantisation in a Magnetic Field'' by Otto Stern and Walter Gerlach in 1922. In this paper, which predated by several years the quantum theory of E. Schroedinger and W. Heisenberg, the authors interpret their experimental results in the light of a semi-classical analysis supplemented with the hypothesis that their atoms were restricted to point either up or down. In modern language, they carried spin-1/2.

When the modern quantum theory appeared in the years 1925-6, spin emerged as a natural consequence of the symmetries of the theory. Since then the Stern Gerlach experiment has taken on a larger meaning, as one of the landmark experiments demonstrating the correctness of quantum mechanics. In its larger role as a key test of quantum mechanics, the Stern Gerlach experiment must be re-interpreted in the light of a theory that was not available in 1922. Some of the properties of a quantum particle are contradicted in the treatment given in the seminal article, and must be revisited from a modern point of view if the results are to be taken as vindicating the modern theory. In our view, this job has not been adequately carried out in standard textbook treatments of the subject, which instead seem to obscure the true complexity of the problem.

We undertake a more thorough job in this research project. The focus of our research is to examine and clarify the behaviour of spin 1/2 particles in an inhomogeneous magnetic field. We have explored the most general solution to the problem within both the classical and quantum mechanical frameworks. We have discovered an interesting subtlety to the problem, with a direct bearing on the interpretation of the Stern-Gerlach experiment. We present both an analytic solution to the classical equations of motion and a numerical solution to the Schroedinger equation for the wave function in various field configurations. We also produce computer animations which help in the visualisation of the dynamics.

In our investigation we consider the most general inhomogeneous magnetic field varying in more than one spatial direction. A practical example of this would be the field which exists inside a magnetic resonance imaging (MRI) apparatus. The MRI device operates by generating a strong inhomogeneous magnetic field in the patient, and then introducing a weak oscillation in the field. When this oscillation is tuned to a certain frequency known as a magnetic resonance, certain atoms inside the patient's tissue rapidly flip their spins between parallel and anti-parallel to the applied magnetic field, inducing a signal in the imaging receiver. In the Stern-Gerlach apparatus there is no oscillation in the applied field. However, we show that the oscillatory motion implied by the classical analysis of Stern and Gerlach would induce weak oscillations in the field sensed by the atom, and that these oscillations are tuned to the magnetic resonance for that atom. If these oscillations actually do cause the atomic spins to flip inside the Stern Gerlach magnet, as this argument would suggest, the key assumption of constant spin that is made in the quantum mechanical solution to the problem is undermined.

Quantum mechanics is the theory that underlies our current understanding of the physical world at the fundamental level. Its predictions have been verified countless times. The point of this research is not to check if quantum mechanics agrees with the Stern Gerlach results, but rather to develop a consistent connection between the theory and the phenomena. If this experiment is a key success of the quantum theory then the connection between the two should be demonstrated clearly and consistently. Our quantum mechanical solution does not rely upon classical arguments which contradict the very interpretation of quantum mechanics that the Stern-Gerlach experiment is meant to verify as do traditional quantum treatments. Furthermore, our quantum solution is of the most general form and applicable to all magnetic fields, including those that exist inside MRI devices.