Chapter 2, Postulates of Quantum Mechanics, points to the complementarity and inseparability of quantum mechanics and spectroscopy. To elucidate the close relationship of spectroscopy and quantum mechanics and to arouse interest in the subject, the collaborative evolution of the theoretical and experimental ideas of the quantization of magnetic moment of particle spin is briefed first. The necessary quantum mechanical background is introduced by discussing the physical and statistical meaning of the postulates. The physical meaning of the observable of a measurement is also explained. The principle of superposition of eigenstates and the time evolution of a state function during system-radiation interactions are analyzed. It is mentioned that a system weakly perturbed by an impulse can often be analyzed by perturbation theory. The chapter ends with a discussion of perturbation approaches to degenerate and nondegenerate systems.
Chapter 2, Postulates of Quantum Mechanics, points to the complementarity and inseparability of quantum mechanics and spectroscopy. To elucidate the close relationship of spectroscopy and quantum mechanics and to arouse interest in the subject, the collaborative evolution of the theoretical and experimental ideas of the quantization of magnetic moment of particle spin is briefed first. The necessary quantum mechanical background is introduced by discussing the physical and statistical meaning of the postulates. The physical meaning of the observable of a measurement is also explained. The principle of superposition of eigenstates and the time evolution of a state function during system-radiation interactions are analyzed. It is mentioned that a system weakly perturbed by an impulse can often be analyzed by perturbation theory. The chapter ends with a discussion of perturbation approaches to degenerate and nondegenerate systems.
This chapter starts with the definition of postulates. Here, in quantum mechanics, the postulates concern the atomic and molecular properties, which are quite far removed from everyday experience. The postulates described here make no claim to those happy mathematical adjectives such as ‘rigorous’, ‘complete’, or irreducible. The ‘proof’ of the postulates is their ability to explain the adequate experimental observations. Here, we have incorporated six postulates. The first postulate opines that ‘the state of quantum mechanical objects is described by a wavefunction’, whereas the second postulate says that ‘every physical observable is expressed by a linear operator’. It also describes how a quantum mechanical operator is constructed. The third postulate gives an idea that the measurement of a physical observable will yield a result, which is one of the Eigen values of the corresponding operator for that observable. The fourth postulate gives how the average of a sequence of measurement is connected with an operator. The fifth postulate is called the time-dependent postulate. Here, it gives a relationship between the time-dependent wave function and an operator, whereas the sixth postulate says that the wave function must be antisymmetric (or symmetric) for the exchange of identical fermions. It must be noted that each and every postulate has been explained clearly. The references, solved problems, and questions on concepts have been given at the end of the chapter.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
This chapter presents a systematic treatment of quantum theory in terms of six groups of postulates on the fundamental properties of quantum systems: postulate on the description of pure states, postulate on the description of observables and their values, postulate on the transition from classical to quantum systems, postulates on the probabilistic relationship between states and observables, postulates on the time evolution of the system and postulate on states after measurement. It extends the basic theory to discuss mixed states, superselection rules and many-particle systems. The chapter presents a brief discussion of conceptual problems in quantum mechanics. A maximum amount of information about the system corresponds to a set of simultaneously measured values of a complete set of discrete observables of the system. Such an amount of information characterises a state of the system. States characterised by such a maximum amount of information about the system are called pure states.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.
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