In this section, we will analyze transitions between energy eigenstates caused by a transient external force, for example, an EM pulse. We will assume that the external force is weak so that we can describe these decays by a version of perturbation theory. This results into very simple expressions for the transition probabilities and rates with universal validity, that is, they apply to all kinds of phenomena from atomic to nuclear and high-energy physics.

It is possible to engineer properties of materials, devices, and systems by changing experimentally available control parameters to optimally approach a specific objective. The following sections demonstrate some potential applications of quantum engineering and show how this may be achieved by the development of efficient physical models combined with optimization algorithms.

In this chapter we present computational Monte Carlo methods to sample from probability distributions, including Bayesian posteriors, that do not permit direct sampling. In doing so, we introduce the basis for Monte Carlo and Markov chain Monte Carlo sampling schemes and delve into specific methods. These include, at first, samplers such as the Metropolis–Hastings algorithms and Gibbs samplers and discuss the interpretation of the output of these samplers including the concept of burn-in and sample correlation. We also discuss more advanced sampling schemes including auxiliary variable samplers, multiplicative random walk samplers, and Hamiltonian Monte Carlo.

This chapter introduces the method for solving time-dependent problems of quantum many-body systems. It includes the pace-keeping DMRG, time-evolving block decimation (TEBD), adaptive time-dependent DMRG, and folded transfer matrix methods. The pace-keeping DMRG, which solves the time-dependent Schrodinger equation, works independently of the dimensionality, nor the model Hamiltonian, with or without impurities. The time-evolving block decimation (TEBD) is more efficient than the pace-keeping DMRG if a one-dimensional Hamiltonian with the nearest-neighboring interactions is studied. The adaptive time-dependent DMRG provides an efficient scheme to implement TEBD with the skill of DMRG. On the other hand, the folded transfer matrix method handles the transfer matrix like TMRG by folding the transfer matrix so that the entanglement entropy along the positive and negative time evolution directions can partially cancel each other. This folding scheme significantly extends the time scale within which a time-dependent problem can be reliably investigated.

This book is a novel synthesis of the philosophy and practice of science, covering its diverse theoretical, metaphysical, logical, philosophical, and practical elements. The process of science is generally taught in its empirical form: what science is, how it works, what it has achieved, and what it might achieve in the future. What is often absent is how to think deeply about science and how to apply its lessons in the pursuit of truth, in other words, knowing how to know. In this volume, David B. Teplow presents illustrative examples of science practice, history and philosophy of science, and sociological aspects of the scientific community, to address commonalities among these disciplines. In doing so, he challenges cherished beliefs and suggests to students, philosophers, and practicing scientists new, epistemically superior, ways of thinking about and doing science.

Several numerical methods used in the study of tensor network renormalization are introduced, including the power, Lanczos, conjugate gradient, Arnoldi methods, and quantum Monte Carlo simulation.

The simplest quantum systems correspond to the smallest nontrivial Hilbert space ${ℂ}^2$. They are called two-level systems. Some physical magnitudes, such as photon polarization or electron spin, are naturally described by the Hilbert space ${ℂ}^2$. However, usually, two-level systems are approximations to more complex systems. Consider, for example, an atom with a lowest-energy state $|0〉$.

Many experiments require the execution of several measurements in a microscopic system. For example, consider a particle A decaying into a pair of particles B and C that move in different directions. Each product particle is detected by a different apparatus at different moments of time. We need a rule that tells us how to incorporate information obtained from the first measurement in order to predict the outcome of the second.

In this chapter we introduce the concept of likelihoods, how to incorporate measurement uncertainty into likelihoods, and the concept of latent variables that arise in describing measurements. We then show how the principle of maximum likelihood is applied to estimate values for unknown parameters and discuss a number of numerical optimization techniques used in obtaining maximum likelihood estimators. These techniques include a discussion of expectation maximization.

One of the most important motives for the development of quantum theory was the need to understand the structure of atoms and molecules, and to account for their emission spectra. Later on, analogous issues were raised for other composite systems, such as nuclei and hadrons. In all cases, the answer requires finding the discrete spectrum of the Hamiltonian $\widehat{H}$ that describes the composite system, that is, solving the eigenvalue equation $\widehat{H}|n〉=E_n|n〉$ for the Hamiltonian.

The tangent space of a tensor network state is a manifold that parameterizes the variational trajectory of local tensors. It leads to a framework to accommodate the time-dependent variational principle (TDVP), which unifies stationary and time-dependent methods for dealing with tensor networks. It also offers an ideal platform for investigating elementary excitations, including nontrivial topological excitations, in a quantum many-body system under the single-mode approximation. This chapter introduces the tangent-space approaches in the variational determination of MPS and PEPS, starting with a general discussion on the properties of the tangent vectors of uniform MPS. It then exemplifies TDVP by applying it to optimize the ground state MPS. Finally, the methods for calculating the excitation spectra in both one and two dimensions are explored and applied to the antiferromagnetic Heisenberg model on the square lattice.

In classical mechanics, the constants of motion of an isolated system are energy, linear momentum, and angular momentum. So far in this book, angular momentum has not been considered. This chapter starts by defining classical angular momentum and then proceeds to find the corresponding quantum operators. Following this, a hydrogenic atom is studied as a prototype application.

The Hamiltonian for $N$ particles subject to mutual two-body interactions

Engineers who design transistors, lasers, and other semiconductor components want to understand and control the cause of resistance to current flow so that they may better optimize device performance. A detailed microscopic understanding of electron motion from one part of a semiconductor to another requires the explicit calculation of electron scattering probability. One would like to know how to predict electron scattering from one state to another – something quantum mechanics can do.