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The Quantum Mechanics Solver 8 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 6 Measuring the Electron Magnetic Moment Anomaly In the framework of the Dirac equation the gyromagnetic factor g of the electron is equal to 2. In other words the ratio between the magnetic moment and the spin of the electron is gq 2m q m where q and m are the charge and the mass of the particle. When one takes into account the interaction of the electron with the quantized electromagnetic field one predicts a value of g slightly different from 2. The purpose of this chapter is to study the measurement of the quantity g 2. 6.1 Spin and Momentum Precession of an Electron in a Magnetic Field Consider an electron of mass m and charge q q 0 placed in a uniform and static magnetic field B directed along the z axis. The Hamiltonian of the electron is H - p qA 2 p B 2m where A is the vector potential A B x r 2 and p is the intrinsic magnetic moment operator of the electron. This magnetic moment is related to the spin operator S by p yS with y 1 a q m. The quantity a is called the magnetic moment anomaly . In the framework of the Dirac equation a 0. Using quantum electrodynamics one predicts at first order in the fine structure constant a a 2n . The velocity operator is V p qA m and we set w qB m. 6.1.1. Verify the following commutation relations Vx H ihw Vy Vy H -ih vx vz H Q . 6.1.2. Consider the three quantities . . . . . . . C1 t Sz vz C2 t SXVX Sy Vy C3 t SxV y - SyVx . y 66 6 Measuring the Electron Magnetic Moment Anomaly Write the time evolution equations for C1 C2 C3. Show that these three equations form a linear differential system with constant coefficients. One will make use of the quantity il a.z. 6.1.3. What is the general form for the evolution of S v 6.1.4. A beam of electrons of velocity v is prepared at time t 0 in a spin state such that one knows the values of 01 0 C2 0 and 63 0 . The beam interacts with the magnetic field B during the time interval 0 T . One neglects the interactions between the electrons of the beam. At time T one measures a quantity .
