Bound states in one-dimensional finite and infinite wells and delta-function potentials, and combinations of these, are obtained; the WKB method for bound states is introduced; the consequences of a parity-invariant potential for the eigenfunctions are derived.

Exchange and permutation operators; symmetrizer and antisymmetrizer; two bosons or two fermions in a central potential; scattering amplitudes of two identical particles in a central potential; the Fermi gas; theory of white dwarf stars; the Thomas-Fermi approximation for many-electron atoms.

Wave-particle duality and the Davisson-Germer experiment are briefly discussed; wave packets are defined and their features, including phase velocity, group velocity, and spreading, are examined; the stationary phase method is presented; free-particle wave functions are introduced, and the equivalence between coordinate- and momentum-space representations of these wave functions is emphasized.

Euler angles and rotation matrices; construction and properties of the rotation matrices; transformation of irreducible tensor operators under rotations; fine-structure of the hydrogen atom; hydrogen atom in a magnetic field: Zeeman and Paschen-Back effects; hyperfine structure of the hydrogen atom; tensor operators; time reversal and irreducible tensor operators.

Explicit solution of the hydrogen-like atom and isotropic harmonic oscillator radial equations by the technique of power series expansion; WKB derivation of the hydrogen-like spectrum; virial theorem; the two-dimensional isotropic harmonic oscillator in plane polar coordinates; the two-body problem and the center-of-mass and relative position and momentum operators.

Construction ofunitary operators inducing space and time translations, and rotations; the anti-unitary operator inducing time reversal; consequences of invariance under a symmetry transformation; periodic potentials and Bloch waves; the Kronig-Penney model; the ammonia molecule and broken parity symmetry; consequences of time reversal invariance on the scattering amplitude of spinless particles; Kramers degeneracy.

The Schroedinger equation for a particle in a potential is introduced and the general properties of its solutions are discussed; the uncertainity relations are derived; the Gram--Schimdt procedure for orthonormalizing a set of independent wave functions is introduced; the time evolution of the expectation values of the position and momentum operatorsfor a particle in a potential and in an electromagnetic field are derived.

In this chapter we consider two examples of the situation when the classicalobservables should be described by a noncommutative (quantum-like)probability space. A possible experimental approach to find quantum-like correlationsfor classical disordered systems is discussed. The interpretation ofnoncommutative probability in experiments with classical systems as a resultof context (complex of experimental physical conditions) dependence ofprobability is considered.

This chapter is devoted to the Bohr complementarity principle.This is one of the basic quantum principles. We dissolve it into separate subprincipleson contextuality, incompatibility, complementary-completeness,and individuality. We emphasize the role of the contexuatlity component. It is not highlighted in the foundational discussions. ByBohr, the outputs of measurements are resulted from the complexinteraction between a system and measurement context, the values ofquantum observables cannot be treated as objective properties of systems.Such Bohr contextuality is more general than joint measurement contextuality(JMC) considered in the discussions on the Bellinequality. JMC is a very special form of the Bohr contextuality. The incompatibility component is always emphasized and often referred as the wave-particle duality. The principle ofinformation complementary-completeness represents Bohr’s claim on completenessof quantum theory. The individuality principle is basic for thenotion of phenomenon used by Bohr to emphasize the individuality and discreteness of outputs of measurements. Individuality plays the crucial role in distinguishing quantum and classical optics entanglements.

We show that for two classical Brownian particles there exists an analog ofcontinuous-variable quantum entanglement: The common probability distributionof the two coordinates and the corresponding coarse-grained velocitiescannot be prepared via mixing of any factorized distributions referring tothe two particles in separate. This is possible for particles which interactedin the past, but do not interact in the present. Three factors are crucial forthe effect: (1) separation of time-scales of coordinate and momentum whichmotivates the definition of coarse-grained velocities; (2) the resulting uncertaintyrelations between the coordinate of the Brownian particle and thechange of its coarse-grained velocity; (3) the fact that the coarse-grained velocity,though pertaining to a single Brownian particle, is defined on a commoncontext of two particles. The Brownian entanglement is a consequenceof a coarse-grained description and disappears for a finer resolution of theBrownian motion. We discuss possibilities of its experimental realizations inexamples of macroscopic Brownian motion.

This chapter presents the basics of the mathematical formalism and methodologyof the prequantum classical statistical field theory (PCSFT). In theBild-conception framework, PCSFT gives an example of acausal theoretical model (CTM) beyond QM, considered as observationalmodel (OM). Generally CTM-OM correspondence is not as straightforwardas in Bell’s model with hidden variables. In PCSFT hidden variables are randomfields fluctuating at spatial and temporal scales which are essentiallyfiner than those approached by the present measurement technology. Thekey element of the PCSFT-QM correspondence is mapping of the complexcovariance operator of a subquantum random field to the density operator.For compound systems, the situation is more complicated. Here PCSFT providestwo descriptions of compound systems with random fields valued intensor vs. Cartesian product of the Hilbert spaces of subsystems. The lattermodel matches representation of compound systems in classical statisticalmechanics. Both approaches are used for measure-theoretic representationof the correlations violating the Bell inequalities.