Exact solutions to the Schrödinger equation for realistic nanoscale systems are beyond reach, hence, different strategies for approximating the solutions are necessary. Perturbation theory relies on a “zero-order Hamiltonian” to express the desired eigenvalues and eigenvectors of “the full Hamiltonian.” We derive working equations for the Rayleigh–Schrödinger perturbation theory, and the validity of the approach is analyzed for a generic two-level system. Applications are given to atoms perturbed by point charges or static fields, and for electrons in quantum wells. An alternative strategy is the variation method, which replaces exact solutions by their projection on a reduced space of “trial functions,” varied to minimize the associated error. Particularly important is the method of linear variation, which can potentially converge to the exact Hamiltonian eigenstates. The mean-field approximation, commonly used for many-particle systems, is derived by optimizing a trial function in the form of product of single-particle functions.

We discuss the breakdown of classical theory in relation to phenomena on the nanoscale. The historical discovery of the wave nature of electrons in the Davisson–Germer Experiment is reviewed. We present the puzzling experimental data and its explanation in terms of particle diffraction, which contradicts classical mechanics. The quantitative success of de Broglie’s formula in associating particle momenta with a wavelength is demonstrated. Analyzing the conditions in which the wave nature of particles becomes apparent, namely, the condition for correspondence between the de Broglie wavelength and the lattice from which the particles are diffracted, we draw some general conclusions. Particularly, by translating to de Broglie wavelengths the particle masses and energy values that are typical to materials and processes on the nanoscales, one immediately realizes that wave properties are expected to be dominant. Quantum mechanics is therefore essential for a proper description of nanoscale phenomena.

We introduce a kinetic theory of electron transport on the nanoscale, formulated in terms of the Fock space of an open many-electron system, and the “second quantization” Hamiltonian. To model a thermal electron reservoir (e.g., a metal electrode), the Fermi–Dirac distribution is derived from the corresponding density operator. A nanoscale system, weakly coupled to the reservoir, is modeled as an impurity. When the Born–Markov and secular approximations are valid, quantum master equations are derived, showing that the impurity equilibrates with the reservoir. To account for charge transport through the impurity, as in atomic point contacts or single molecule junctions, the master equations are generalized for cases of an impurity coupled to different reservoirs at different chemical potentials/temperatures. In these cases, we show that the system reaches a nonequilibrium steady state, where current flows through the impurity. Analytic expressions are derived for this steady state in simple models.

The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.

Typically, we are interested in a small system (a few particles, or some region in space), entangled with its surroundings. Exact equations (Dyson or Nakajima–Zwanzig) for the reduced system dynamics are readily derived using suitable projection operators. They are rarely solvable, however, since full account is taken for mutual influences between the system and its environment. Nevertheless, for weak mutual coupling, environment-induced dynamics in the (small) system can be much slower than system-induced dynamics in the (large) environment. This justifies the Born–Markov approximation, leading to closed equations for the system. In Hilbert space, we demonstrate the emergence of irreversible dynamics and exponential decay for pure sates. In Liouville space, we derive the Redfield equation. Invoking the secular (or, rotating-wave) approximation, we derive Pauli’s master equations, which properly account for relaxation to equilibrium. Rates of spontaneous emission, coherence transfer (Bloch equations), and pure dephasing are derived and analyzed for a dissipative qubit.

The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.

According to quantum mechanics, the information with respect to any measurement on a physical system is contained in a mathematical object, the wave function. In this chapter we become familiar with the mathematical objects that represent the measured properties themselves, namely the quantum mechanical operators. We start from a brief introduction into operators and their properties, emphasizing linear operators, and noncommuting operators. Then we introduce the canonical position and momentum operators. Defining functions of operators, we derive different quantum mechanical operators that correspond to different physical observables, including angular momentum, kinetic energy, and the scalar potential energy. Finally, we introduce the quantum mechanical total energy operator (the Hamiltonian) and demonstrate its explicit generic form for nanoscale building blocks such as atoms and molecules.

In many cases, measurements are performed on “mixed” ensembles of realizations of a system, which cannot be associated with a single vector in its Hilbert space. In these cases the state of the system is represented by a proper “density operator.” It is instructive to associate density operators with state vectors in a vector space, termed Liouville’s space, where the Schrödinger equation is reformulated as the Liouville–Von Neumann equation. An important consequence of this equation is that the density operator of a system at equilibrium must commute with its Hamiltonian. Namely, the matrix representation of the density operator in the basis of Hamiltonian eigenstates is diagonal, where the diagonal elements are the relative populations of the system’s Hamiltonian eigenstates. The equilibrium populations are revealed by maximizing the (Von Neumann) entropy, subject to given constraints. We derive the equilibrium density operator explicitly for the cases of canonical and grand canonical ensembles.

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.

Energy quantization in nanoscale materials is manifested in a “blue-shift” of the emission spectrum of nanoparticles of decreasing size. The phenomenon is known as the “quantum size effect,” namely, the increasing gaps between energy levels for a spatially confined particle. The effect is demonstrated by solving the one-dimensional Schrödinger equation for the “particle in a box” model. The confining potential translates into boundary conditions, which result in energy quantization, where the corresponding standing wave solutions demonstrate remarkable differences from the classical description. Different energy levels are obtained by changing boundary conditions to periodic, for a “particle on a ring,” where the phenomenon energy level degeneracy is introduced. Extending the discussion to multidimensional “boxes” enables one to analyze the energy spectrum and the density of states nanostructures including quantum dots, wires, and wells, with references to the devices based on a two-dimensional electron gas, and to quantized conductance through point contacts.

According to the postulates of quantum mechanics, the state of a system is associated with a wave function that contains any measurable information on the system at any time. In this chapter we become familiar with wave functions and how they represent the position of particles within the system. Within the realm of quantum mechanics, the position of particles is not deterministic. It is defined by a probability distribution. The wave function is a position-dependent complex-valued amplitude, whose absolute value squared is identified with the probability density for locating the particle in the position space. This identification of the wave function with a probability amplitude imposes some limitation. Particularly, for a closed system in which the particles are bound, the wave function must be proper (square integrable) and normalizable. These properties are discussed and demonstrated for different coordinate systems.

Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.

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.

The rates of charge transfer and energy transfer are essential for understanding nanoscale phenomena and processes. Here we introduce the general conditions for the emergence of rate processes in quantum mechanics. We refer to the generic scenario in which a transition is induced between eigenstates of a given Hamiltonian by a weak perturbation. Analysis reveals that when the initial and final states are pure states, the transition rate is time-dependent and fails to reach a finite constant value. If, however, the final state is a mixed ensemble that is sufficiently wide and dense in energy, a rate constant emerges, given by Fermi’s Golden Rule (FGR). When the initial state is also mixed, for example, a thermal equilibrium state, a thermal rate constant is formulated in terms of transition rates between specific eigenstates, summed over the final eigenstates, and averaging over the thermal distribution of the initial eigenstates.

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.

The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.