Abstract In this chapter we introduce the quantum mechanical description of the state of spin ensembles and of their time evolution The relation between the density operator and the magnetization vector of ensembles of isolated spins S = 1/2 is explained. We derive the equation of motion of the density operator from the Schrodinger equation and discuss how the evolution of spin systems during pulse EPR experiments can be mathematically described.
Abstract In this chapter we introduce the quantum mechanical description of the state of spin ensembles and of their time evolution The relation between the density operator and the magnetization vector of ensembles of isolated spins S = 1/2 is explained. We derive the equation of motion of the density operator from the Schrodinger equation and discuss how the evolution of spin systems during pulse EPR experiments can be mathematically described.
This chapter outlines the quantum ideas that underpin magnetic resonance and explains their link to classical physics. It discusses some of the essential concepts governing the quantum behaviour of ensembles of nuclear spins, including the nature of quantum states, the eigenvalue equation and observables, the measurement process, dynamics, and the Schroedinger equation. A detailed description of angular momentum is given along with discussion of the role of symmetry, rotation operators, and both product and total angular momentum representations. Statistical ensembles of spins are introduced and the density matrix and Liouville formalism outlined, along with the various tensor bases for the description of spin systems, both independent and coupled. The spin Hamiltonian is described in detail and the evolution of spin systems under various Hamiltonian terms is discussed using the density matrix description, along with suitable recession diagrams.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
The time-dependent Schrödinger equation forms the chapter's starting point. For a time-independent Hamiltonian, a quantum state may be time evolved with the standard, unitary time evolution operator. After obtaining this operator, an algorithm is provided by which a state may be time evolved, and measurement probabilities determined. A spin 1/2 example illustrates time evolution concretely. All this makes close contact with the discussion of unitary operators in Chapter 10. The time dependence of expectation values forms the next topic, followed by the concept of a constant of the motion, which is carefully contrasted with that of a stationary state. The energy-time uncertainty relations, which rest on a quite different conceptual footing than the ‘standard’ uncertainty relations, occupy the rest of the chapter. An example from spin 1/2 illustrates the energy-time uncertainty relations.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles