This chapter starts out by introducing the energy eigenvalue equation – the time-independent Schrödinger equation. Firstly, the notion of energy quantization is introduced by semi-analytical means. A particle is confined within a rectangular well. It is seen, eventually, that the corresponding solutions to the time-independent Schrödinger equation can only exist for a few, specific energies. This phenomenon is also seen for other physical quantities, such as angular momentum and charge. Next, the situation in which the potential experienced by a quantum particle is periodic is given particular attention. It may be studied numerically by minor adjustments of the framework already developed. The case of a periodic potential is important as, to a large extent, it forms the basis for understanding solid state physics. Since the Hamiltonian is Hermitian, its eigenvectors form an orthonormal set in which any state or wave function may be expanded. This is exploited in the last part of the chapter, which is dedicated to determining ground states – the energy eigenstate of minimal energy. This is done in two ways – by using what is called the variational principle and by so-called propagation in imaginary time. These methods are implemented for several examples and compared to full solutions.

This chapter exploresthe Fermi–Pasta–Ulam–Tsingou (FPUT) problem in the context of a one-dimensional chain of interacting point masses. We start by modelling the system as a chain of masses interacting through a force dependent on their relative displacements. Next, we simplify this system to harmonic oscillators under a linear force dependency, further developing it to describe a wave-like behaviour. The chapter discusses dispersion relations and the impact of boundary conditions leading to discretisation of allowed wave modes. A non-linear, second-order interaction is then included, complicating the system’s dynamics and necessitating the use of numerical methods for its solution. We then track the system’s evolution in a multidimensional phase space, leading to observations of seemingly chaotic motion with emerging periodicity. Energy conservation and its flow through the system are crucial aspects of the analysis. A detailed numerical procedure is provided, involving solution of initial value problems, mode projections, and energy computations to explore the complex behaviour inherent in the FPUT problem.

Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.

Newton’s laws and consideration of units are used to present and discuss the mass on a spring as an example of a harmonic oscillator, a mechanical oscillator with a sinusoidal time dependence. Both the transient solution, where the oscillator is started away from equilibrium and left on its own, and the driven solutions, where a sinusoidal driving force is applied, are presented. The quality factor, Q, is introduced, which characterizes the relative amount of damping forces present, such as those of friction or air drag. The quality factor is related to the number of oscillations that are made when left on its own and to the excitation bandwidth, the range of frequencies over which resonance is observed. Musically relevant examples include the ocarina, tuning fork, some speaker enclosures, and the phenomenon of sympathetic resonance.

The motion of a floating body in waves obeys a damped mass spring equation. The respective roles of mass, damping and stiffness need to be clearly understood. Harmonic and non-harmonic excitations are considered, together with different forms of damping: linear, quadratic, Coulomb.

We discuss the definition of the Schrödinger operator on Euclidean space as a self-adjoint operator and discuss basic properties of its spectrum, including criteria for discreteness and finiteness of its negative spectrum. We analyze in detail the classical examples of the harmonic oscillator, the Coulomb Hamiltonian and the Pöschl–Teller potential, which can be solved using a commutation method. Returning to general potentials, we use Dirichlet–Neumann bracketing to prove Weyl asymptotics for the number and Riesz means of negative eigenvalues in the strong coupling constant limit. These asymptotic results are complemented by the nonasymptotic results of Lieb–Thirring, Cwikel–Lieb–Rozenblum, and Weidl. We present a unified method of proof of these bounds, based on Sobolev inequalities and the Besicovitch covering lemma. As an application of these bounds, we extend Weyl asymptotics to a large class of potentials.

We solve the quantum mechanical harmonic oscillator problem using an operator approach. We define the lowering and raising operators. We use the quantum mechanical harmonic oscillator to review the fundamental ideas of quantum mechanics.We study some examples of time dependence in the harmonic oscillator including the coherent state. We apply the quantum mechanical harmonic oscillator to the study of the vibrations of the nuclei of molecules.

We learn how to use perturbation theory to solve more realistic problems that do not admit exact solutions. We learn degenerate and nondegenerate perturbation theory and apply them to a variety of problems, including spin magnetic moments in magnetic fields and the Stark effect in hydrogen.

This chapter provides a self-contained introduction to the basic aspects of Quantum Mechanics, focusing on what is must for Quantum Field Theory. The notions of state space, unitary operators, self-adjoint operators, and projective representation are covered as well as Heisenberg’s uncertainty principle. A complete proof of Stone’s theorem is given, but the spectral theory is covered only at the heuristic level. We provide an introduction to Dirac’s formalism, which is almost universally used in physics literature. The time-evolution is described in both the Schrödinger and the Heisenberg picture. A complete treatment of the harmonic oscillator, providing an introduction to the fundamental idea of creation and annihilation operators concludes the chapter.