Given the description of a quantum state in terms of Hilbert space vectors, physical magnitudes (Heisenberg’s matrices) correspond to linear operators on the Hilbert space. A linear operator (or simply, operator) is a linear map of a Hilbert space to itself.

Science is distinguished from other endeavors by the scientific method, which starts with curiosity and leads sequentially from hypothesis to experimental testing to hypothesis revision, and finally, to knowledge. This chapter traces the development of the scientific method from ancient Egypt, Greece, the Islamic world, Europe (beginning in the Middle Ages), and the modern world (eighteenth to twenty-first century). It shows how the method became increasingly rigorous and precise through codification of its practice and the use of statistics in data analysis. The contributions of philosophy to the method and its possible senescence, in the light of data-driven science, also are discussed.

In this chapter we introduce the clustering problem and use it to motivate mixture models. We start by describing clustering in a frequentist paradigm and introduce the relevant likelihoods and latent variables. We then discuss properties of the likelihoods including invariance with respect to label swapping. Finally, we expand this discussion to describe clustering and mixture models more generally within a Bayesian paradigm. This allows us to introduce Dirichlet priors used in inferring the weight we ascribe to each cluster component from which the data are drawn. Finally, we describe the infinite mixture model and Dirichlet process priors within the Bayesian nonparametric paradigm, appropriate for the analysis of uncharacterized data that may contain an unspecified number of clusters.

In this chapter we present dynamical systems and their probabilistic description. We distinguish between system descriptions with discrete and continuous state-spaces as well as discrete and continuous time. We formulate examples of statistical models including Markov models, Markov jump processes, and stochastic differential equations. In doing so, we describe fundamental equations governing the evolution of the probability of dynamical systems. These equations include the master equation, Langevin equation, and Fokker–Plank equation. We also present sampling methods to simulate realizations of a stochastic dynamical process such as the Gillespie algorithm. We end with case studies relevant to chemistry and physics.

This chapter extends DMRG from real space to an arbitrary basis space in which each basis state, such as a momentum eigenstate or a molecular orbital, serves as an effective lattice site. Unlike in real space, the interaction potentials become nonlocal and off-diagonal in an arbitrary basis representation. To solve this nonlocal problem, one should optimize the order of basis states and introduce the so-called complementary operators to minimize the number of operators whose matrix elements must be computed and stored. We illustrate the momentum-space DMRG using the Hubbard model and discuss its application in other interacting fermion models. Finally, we introduce a DMRG scheme for optimizing the single-particle basis states and their order simultaneously in a more general basis space without momentum conservation.