Time-Dependent Quantum Mechanics of Two-Level Systems

James P Lavine

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813272606
• Topic: Naturwissenschaften
• Subtopic: Quantentheorie

With both industrial and teaching experience, the author explains the effects of time dependence in systems with two energy levels. The book starts with time-independent interactions and goes on to treat interactions with time-dependent electric and magnetic fields. Complete derivations are presented for each case, so the reader understands how the solutions are found. Both closed-form and numerical solutions are treated, and the calculations are compared with experimental data from the literature. Numerous plots are provided to show how the solutions depend on the parameters of the interactions.

The book builds upon an undergraduate course in quantum mechanics and is useful for readers interested in magnetic resonance and quantum optics. In addition, this book is ideal for self-study by students or researchers starting on two-level systems. The detailed derivations and plots should ease readers into the study of two-level systems in a wide variety of settings.

Contents:

• Preface
• Spin Precession and Rate Equations
• Two-Level Systems with a Time-Independent Interaction
• Two-Level Systems with a Time-Dependent Interaction
• Applications and Examples of Two-Level Systems
• The Density Matrix and the Relaxation of Two-Level Systems
• The Second-Order Correlation Function for Two-Level Systems
• Appendices:
  • Physical Constants
  • The Electromagnetic Field
  • Diagonalization of a Matrix in Chapter 2
  • The Time Evolution Operator and the Sudden Approximation
  • Beyond the Rotating Wave Approximation
  • The Second-Order Correlation Function for a Two-Level System
• References
• Index

Readership: This is an undergraduate textbook in quantum mechanics suitable for students in physics, chemistry, optics, materials science, and electrical engineering.
Key Features:

• The book starts at entry level and doesn't require knowledge in quantum mechanics
• Detailed derivations of the solutions and plots of the results
• Treats a wide range of time-dependent two-level problems. Closed-form solutions are provided for most cases; experimental data are included and compared to calculations

Chapter 1

Spin Precession and Rate Equations

Time dependence involves changes in the occupation of the states of a physical system. For example, the electron in a hydrogen atom may be excited from a 1s state into a 2p state due to a perturbation such as an electromagnetic field. (The states of the hydrogen atom are derived in Quantum Mechanics texts such as Konishi and Paffuti (2009, Chap. 6).) Generally, we say a transition has occurred when the physical system changes its state. It is simplest to view this as a change from the system being in state i to the system being in state j. This chapter starts with basic ideas on time dependence in Quantum Mechanics and the time evolution operator. These are applied to spin precession of a particle. Then the Einstein coefficients and rate equations are introduced. The following chapters are concerned with calculating the probability p_i(t) for the occupation of state i of a system as a function of the time t. These predictions are compared with experiments whenever possible.

Section 1.1 briefly states what we explore and Sec. 1.2 introduces the Schrödinger Equation and the time evolution operator. These are applied to the precession of the spin angular momentum for a spin-

particle in a constant magnetic field in Sec. 1.3. This example also serves as a review of the Pauli matrices. Section 1.4 presents the Einstein coefficients and the relationships between absorption, stimulated emission and spontaneous emission of electromagnetic radiation. Section 1.5 discusses rate equations, which are often used to describe transitions within a physical system. This chapter ends with two Rambles in Sec. 1.6. These touch on the wave function of the neutron and on the spin precession of the muon. Data are presented for both topics and compared with the theory developed in this chapter. Appendix 1 collects the values of useful physical constants, and App. 2 covers the electromagnetic field and includes some relationships that help us understand experiments.

1.1.Experiments and Time Dependence

Experimental probes take a variety of forms. We start with a beam of photons as an example of a probe of atoms by electromagnetic radiation. We assume a set of identical physical systems, each with its state 2 having a higher energy than its state 1. The photon beam falls on our sample and we measure the incident intensity and the transmitted intensity. From these we hope to deduce, for example, information on the absorption probability of state 1 versus the photon wavelength. Alternatively, we may observe our sample and detect the emission of radiation when a physical system makes a transition from state 2 to state 1. In both cases, we learn about the occupation probabilities for states 1 and 2 and the energy difference between them.

We let p₁(t) and p₂(t) be the occupation probabilities of states 1 and 2, respectively, as a function of the time t. These probabilities may be obtained from measurements on a sample with an ensemble of phy...