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This chapter discusses abstract linear space of state vectors. The wave mechanics presented in the previous chapter is easily generalized for use in all the applications of quantum mechanics explained in this book. In particular, to take account of spin, one just replaces the wave function with a set of functions, one for each possible choice of the quantum numbers of the z components of the spins of the particles. However, as the chapter shows, it is easy to adapt the wave mechanics formalism to the more general scheme that represents the states of a system as elements of an abstract linear space rather than a space of wave functions. This approach has the virtue that one can explicitly see the logic of the generalization of the wave function to take account of spin, and this is the road to other generalizations, like quantum field theory.

This chapter introduces the matrix-mechanics formulation of quantum mechanics, emphasizing both calculational techniques and conceptual understanding. Parallels between matrix mechanics and ordinary vectors and matrices are extensively utilized. Starting with the representation of ordinary vectors as rows or columns of numbers, the scalar product is discussed, followed by the transformation of vectors by matrices, as illustrated by rotations. The vector representation of quantumstates, the inner product of two such states, and the matrix representation of operators are then introduced. The simple forms assumed in matrix mechanics by a basis state, and by an operator, when either is written in its eigenbasis, are discussed, as are the specific forms of adjoint, Hermitian, and unitary operators. The chapter concludes with a brief exposition of eigenvalue equations in matrix mechanics.

This chapter reviews measurement theory in quantum mechanics. The measurement prescription in quantum mechanics can be stated in a few lines and has found an enormous range of applications, in all of which it has proved to be consistent with logic and experimental tests. However, the implications seem so bizarre that people such as Albert Einstein and Eugene Wigner have argued that the theory cannot be physically complete as its stands. The chapter then extends the prescription to the case where the state vector is not known. It also discusses some of the “paradoxes” of quantum mechanics. Finally, the chapter presents Bell's theorem, which shows that there cannot be a local underlying deterministic theory for which quantum mechanics plays the role of a statistical approximation.

Particle learning provides a simulation‐based approach to sequential Bayesian computation. To sample from a posterior distribution of interest we use an essential state vector together with a predictive distribution and propagation rule to build a resampling‐sampling framework. Predictive inference and sequential Bayes factors are a direct by‐product. Our approach provides a simple yet powerful framework for the construction of sequential posterior sampling strategies for a variety of commonly used models.

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out on the ensemble, it will split into a number of sub-ensembles, each sub-ensemble being conditioned on a particular outcome of the measurements. The mathematical description of the collection of sub-ensembles thus created leads to probability distributions on projective Hilbert space. The chapter develops an appropriate mathematical framework which enables the general formulation of such a distribution, and leads to the concepts of stochastic state vectors and stochastic density matrices. These concepts are required in later chapters to construct appropriate stochastic differential equations describing the continuous monitoring of open quantum systems.

This chapter discusses maximum likelihood estimation of parameters both for the case where the distribution of the initial state vector is known and for the case where at least some elements of the vector are diffuse or are treated as fixed and unknown. For the linear Gaussian model it shows that the likelihood can be calculated by a routine application of the Kalman filter, even when the initial state vector is fully or partially diffuse. It details the computation of the likelihood when the univariate treatment of multivariate observations is adopted. It considers how the loglikelihood can be maximised by means of iterative numerical procedures. An important part in this process is played by the score vector. The chapter shows how this is calculated, both for the case where the initial state vector has a known distribution and for the diffuse case. A useful device for maximisation of the loglikelihood in some cases, particularly in the early stages of maximisation, is the EM algorithm; details are provided for the linear Gaussian model. The chapter also considers biases in estimates due to errors in parameter estimation and ends with a discussion of some questions of goodness-of-fit and diagnostic checks.

Computational algorithms in state space analyses are mainly based on recursions, that is, formulae in which the value at time t + 1 is calculated from earlier values for t, t − 1, …, 1. This chapter deals with the question of how these recursions are started up at the beginning of the series, a process called initialisation. It provides a general treatment in which some elements of the initial state vector have known distributions while others are diffuse, that is, treated as random variables with infinite variance, or are treated as unknown constants to be estimated by maximum likelihood. The discussions cover the exact initial Kalman filter; exact initial state smoothing; exact initial disturbance smoothing; exact initial simulation smoothing; examples of initial conditions for some models; and augmented Kalman filter and smoother.

The evolution of quantum mechanics through the 1920s was profoundly messy. Some physicists believed that it was necessary to throw out much of the conceptual baggage that early quantum mechanics tended to carry around with it and re-establish the theory on much firmer ground. It was at this critical stage that the search for deeper insights into the underlying reality was set aside in favour of mathematical expediency. All the conceptual problems appeared to be coming from the wavefunctions. But whatever was to replace them needed to retain all the properties and relationships that had so far been discovered. Dirac and von Neumann chose to derive a new quantum formalism by replacing the wavefunctions with state vectors operating in an abstract Hilbert space, and formally embedding all the most important definitions and relations within a system of axioms.

In the previous six chapters, we focused on understanding the basic eigenstates of various Hamiltonians by solving the time independent Schrödinger equation. While the Hamiltonians seemed relatively simple, the solutions are frequently used. However, in practice, most of the important problems today do not make extensive use of the spatial solutions except when a specific number is needed. The focus on most problems today makes extensive use of operators where their behavior is governed by the new design rules that we must follow to do anything novel with quantum systems. The design rules are sometimes referred to as the postulates. Postulates are mathematical statements of specific rules. We will present five postulates in this chapter and then in later chapters augment them with three more. In the context of this math, we will continue to work with functions like ψ(r,t), but we will introduce Dirac notation such as ∣ψ(t)〉. The remaining chapters in this text work almost exclusively in Dirac notation and with operators.