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The theory of quantum propagator and time--dependent integrals of motion in quantum optics is reviewed as well as the properties of Wigner function, Q--function, and coherent state representation. Propagators and wave functions of a free particle, harmonic oscillator, and the oscillator with varying frequency are studied using time--dependent linear in position and momentum integrals of motion. Such nonclassical states of light (of quantum systems) as squeezed states, correlated states, even and odd coherent states (Schr\"odinger cat states) are considered. Photon distribution functions of Schr\"odinger cat male and female states are given, and the photon distribution function of squeezed vacuum is derived using the theory of the oscillator with varying parameters. Properties of multivariable Hermite polynomials used for the description of the multimode squeezed and correlated light and polymode Schr\"odinger cats are studied.

The theory of quantum propagator and time--dependent integrals of motion in quantum optics is reviewed as well as the properties of Wigner function, Q--function, and coherent state representation. Propagators and wave functions of a free particle, harmonic oscillator, and the oscillator with varying frequency are studied using time--dependent linear in position and momentum integrals of motion. Such nonclassical states of light (of quantum systems) as squeezed states, correlated states, even and odd coherent states (Schr\"odinger cat states) are considered. Photon distribution functions of Schr\"odinger cat male and female states are given, and the photon distribution function of squeezed vacuum is derived using the theory of the oscillator with varying parameters. Properties of multivariable Hermite polynomials used for the description of the multimode squeezed and correlated light and polymode Schr\"odinger cats are studied.

The theory of quantum propagator and time--dependent integrals of motion in quantum optics is reviewed as well as the properties of Wigner function, Q--function, and coherent state representation. Propagators and wave functions of a free particle, harmonic oscillator, and the oscillator with varying frequency are studied using time--dependent linear in position and momentum integrals of motion. Such nonclassical states of light (of quantum systems) as squeezed states, correlated states, even and odd coherent states (Schr\"odinger cat states) are considered. Photon distribution functions of Schr\"odinger cat male and female states are given, and the photon distribution function of squeezed vacuum is derived using the theory of the oscillator with varying parameters. Properties of multivariable Hermite polynomials used for the description of the multimode squeezed and correlated light and polymode Schr\"odinger cats are studied.

The education system for students in physics suffers (worldwide) from the absence of a deep course in probability and randomness. This is the real problem for students interested in quantum information theory, quantum optics, and quantum foundations. Here the primitive treatment of probability and randomness may lead to deep misunderstandings of theory and wrong interpretations of experimental results. Since during my visits (in 2013 and 2014) to IQOQI a number of students (experimenters!) asked me permanently about foundational problems of probability and randomness, especially inter-relation between classical and quantum structures, this year I gave two lectures on these problems. Surprisingly the interest of experiment-oriented students to mathematical peculiarities was very high. This (as well as permanent reminding of prof. Zeilinger) motivated me to write a text based on these lectures which were originally presented in the traditional black-board form. I hope that this might be useful for students from IQOQI as well as other young physicists.