Basic Concepts and Axioms

Abstract

The postulates of quantum mechanics are formulated using the mathematical tools of the preceding chapter. First, the axioms related to the quantum kinematics are summarized, dealing with a variety and physical meaning of quantum states at the specified time. They include alternative definitions and interpretations of the wave functions of microobjects as amplitudes of the particle probability distributions in the configuration or momentum spaces. As an illustrative example the electron densities are then discussed. The superposition principle is formulated, and the symmetry implications of indistinguishability of identical particles in quantum mechanics are examined. The links between the quantum states and outcomes of the physical measurements are then surveyed and the physical observables are attributed to quantum mechanical operators, linear and Hermitian, and their specific forms in the position and momentum representations are introduced. The eigenvalues of the quantum mechanical operator are postulated to determine a variety of all possible results of a single experiment measuring the physical property the operator represents, while the operator expectation value represents the average value of this quantity in a very large number of repeated measurements performed on the system in the same quantum state. The eigenstates of the quantum mechanical operator are shown to correspond to the sharply specified value of the physical property under consideration, while other quantum states exhibit distributions of its allowed eigenvalues. The statistical mixtures of quantum states are defined in terms of the density operator and the ensemble averages of physical observables in such mixed states are examined. The simultaneous sharp measurement of several physical observables is linked to the mutual commutation of their operators and the quantum mechanical formulation of the general Principle of Indeterminacy is given. Properties of the electron angular momentum and spin operators are examined. In the dynamical development, the pictures of time evolution in quantum mechanics are introduced through the alternative time-dependent unitary transformations of the state vectors/operators. The Schrodinger equation is explored in some detail, with the emphasis placed upon the stationary states, time dependence of expectation values, conservation laws, the probability current, and continuity equation. The correspondence between the quantum and classical dynamics is established through the Ehrenfest principle. Finally, the rudiments of the Heisenberg and interaction pictures of quantum dynamics are briefly summarized.