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Oxford University Press eBooks

The Quantum Postulates

Gary E. Bowman

Published Nov 1, 2007• 0 citations
Publisher

Abstract

This chapter discusses the concept of a postulate and develops the basic structure of quantum mechanics using three postulates. Postulate 1 introduces the concept of a quantum state as a solution of the time-independent Schrödinger equation. Postulate 2 introduces observables, which are represented by Hermitian operators; the possible measurement values of an observable are the eigenvalues of the corresponding Hermitian operator. Postulate 3 introduces the concept of a superposition, and the association of the probability for obtaining some measurement result with the complex square of an expansion coefficient in a superposition. In discussing Postulate 3, state normalization, the distinction between discrete and continuous eigenvalues, and the Dirac delta are also introduced.

DOI
10.1093/acprof:oso/9780199228928.003.0002

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Cambridge University Press eBooks•May 11, 2023•0 citations
This chapter discusses the concept of a postulate and develops the basic structure of quantum mechanics using three postulates. Postulate 1 introduces the concept of a quantum state as a solution of the time-independent Schrödinger equation. Postulate 2 introduces observables, which are represented by Hermitian operators; the possible measurement values of an observable are the eigenvalues of the corresponding Hermitian operator. Postulate 3 introduces the concept of a superposition, and the association of the probability for obtaining some measurement result with the complex square of an expansion coefficient in a superposition. In discussing Postulate 3, state normalization, the distinction between discrete and continuous eigenvalues, and the Dirac delta are also introduced.
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
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.
This chapter reviews the fundamental assumptions underlying quantum mechanics, while simultaneously seeing how they can be realized in the language of MATLAB, which we will be using to express and explore bonding concepts. For every measurable quantity there is an operator in quantum mechanics. The only possible values that can be measured for a property are given by the eigenvalues of its corresponding operator. Such oscillations represent the usual difficulty of representing a sharp discontinuous function with harmonic functions, such as sine and cosine functions. The time-dependent Schrodinger equation is what gives the eigenfunctions of the Hamiltonian operator such an important place in quantum mechanics. The postulate is fundamental to defining the relationship between a quantum mechanical system and the person observing it. Observations cannot be made on a system without disturbing it, with the sole exception being the case when a system is already in an eigenstate of the property being measured.
The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean. BASIC FEATURES OF QUANTUM OBSERVABLES This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.