Chapter 7: Time Dependence of Amplitudes

Time Dependence of Amplitudes in quantum mechanics, Volume III of The Feynman Lectures on Physics, explores the crucial concept of how probability amplitudes change over time for fundamental particles. The discussion begins by defining stationary states for an atom at rest, emphasizing that while such an atom has a definite energy (E), it possesses no translational motion, momentum, or kinetic energy. The amplitude for a particle in such a state varies only temporally, following the mathematical relationship based on the exponential factor related to energy (E) and the reduced Planck constant (h-bar) over time (t). The text then extends this concept to particles in uniform motion, applying relativistic principles. It introduces the necessary transformations between reference frames, showing how the probability amplitude varies in space and time in a moving system. This analysis leads to defining crucial relationships involving the particle's momentum (p) and energy (E) in terms of wave number (k) and frequency (omega), ultimately demonstrating that the classical velocity is equivalent to the quantum mechanical group velocity (vg), which is defined by the derivative of the frequency (omega) with respect to the wave number (k). Next, the chapter incorporates the effects of potential energy (V), affirming the general principle of energy conservation in the amplitudes description, where the total energy is the sum of internal, kinetic, and potential energies. A significant quantum effect is detailed: when a particle encounters a potential barrier where the potential energy is (greater than) its total energy, the particle's amplitude does not vanish immediately but instead decays exponentially through the barrier. This phenomenon, known as quantum mechanical penetration or tunneling, is illustrated using the example of alpha decay in uranium nuclei, where the probability amplitude for finding the particle exponentially decreases inside the barrier. Finally, the principles of amplitude dependence are applied to a particle with spin one-half, specifically analyzing the disintegration of a muon in a magnetic field. By tracking the time evolution of the amplitudes for spin-up and spin-down states, the analysis reveals that the spin effectively undergoes a continuous precession at a characteristic frequency (omega-p), which is determined by the magnetic moment (mu), the magnetic field (B), and the reduced Planck constant (h-bar), providing an important description of how particle spin behaves over time in external fields.