This chapter introduces probabilistic methods in quantum mechanics, focusing on photon behavior, demonstrating their indivisibility through experiments with single-photon detectors and analyzing probabilistic reflection and transmission at an air-to-glass interface. It introduces phasors to represent quantum events, stating that event probabilities are the squares of phasor amplitudes, and applies this method to analyze partial reflection and transmission, while introducing Heisenberg's uncertainty principle through probabilistic quantum analysis. The chapter explores the quantum mechanical interpretation of photon reflection and transmission at optical interfaces, introducing probability phasors where event amplitudes relate to the square root of probabilities, and discusses partial reflection from optical windows, calculating probabilities of various paths and summing their phasors. It concludes with a quantum interpretation of the law of reflection, showing that classical reflection emerges from summing probability phasors for all possible paths. Additionally, the chapter examines quantum interpretations of classical optical phenomena, demonstrating that straight-line paths are most probable due to aligned probability phasors, and extends this analysis to refraction, slit experiments, and light focusing through lenses, all within a quantum framework accessible through algebra and trigonometry.
This chapter introduces probabilistic methods in quantum mechanics, focusing on photon behavior, demonstrating their indivisibility through experiments with single-photon detectors and analyzing probabilistic reflection and transmission at an air-to-glass interface. It introduces phasors to represent quantum events, stating that event probabilities are the squares of phasor amplitudes, and applies this method to analyze partial reflection and transmission, while introducing Heisenberg's uncertainty principle through probabilistic quantum analysis. The chapter explores the quantum mechanical interpretation of photon reflection and transmission at optical interfaces, introducing probability phasors where event amplitudes relate to the square root of probabilities, and discusses partial reflection from optical windows, calculating probabilities of various paths and summing their phasors. It concludes with a quantum interpretation of the law of reflection, showing that classical reflection emerges from summing probability phasors for all possible paths. Additionally, the chapter examines quantum interpretations of classical optical phenomena, demonstrating that straight-line paths are most probable due to aligned probability phasors, and extends this analysis to refraction, slit experiments, and light focusing through lenses, all within a quantum framework accessible through algebra and trigonometry.
This chapter introduces quantum optics, focusing on wave-particle duality, superposition, entanglement, and quantum teleportation, while employing algebra, trigonometry, and physical reasoning to make quantum mechanics accessible without calculus. It systematically builds from classical optics to historical experiments revealing quantum behavior, using examples and practice problems designed for technician-level education, and prepares students for further study in quantum information science. The chapter explores fundamental properties of light waves, including coherence, polarization, and reflection, with coherent light exhibiting organized interference patterns and incoherent light lacking phase consistency. Polarization is discussed, distinguishing between unpolarized and polarized light, and polarizers are introduced as tools to filter light based on its polarization state. The law of reflection is explained, detailing how light bounces off surfaces based on the angle of incidence and media properties. The chapter also discusses reflection and refraction at media interfaces, introducing Fresnel equations for calculating reflectance for perpendicular and parallel polarized light, simplifying these for normal incidence, and presenting an example calculation for an air–glass interface.
This chapter introduces core quantum mechanics concepts, focusing on Heisenberg's uncertainty principle and Schrödinger's equation, which describe quantum particles' wave–particle duality and probabilistic nature. The uncertainty principle and Schrödinger's equation are fundamental postulates validated by experiments, with the latter governing quantum behavior. The chapter explores wave functions, superposition, and entanglement, using light polarization to contrast classical and quantum interpretations, demonstrating that single photons exhibit probabilistic outcomes when interacting with polarizers. Additionally, it discusses polarizers and polarizing beam splitters, analyzing light propagation through these components using the law of Malus and comparing classical and quantum interpretations. The chapter concludes by highlighting the implications of these findings on photon polarization and measurement, providing a foundation for understanding quantum technologies.
Abstract Quantum mechanics is a physical theory based on statistical laws and can merely predict probabilities for the measurement outcomes, suggesting that it is an average description. Due to the lack of all the details of microscopic physical objects, only probabilistic predictions can be made for physical phenomena. In this paper, we discuss the root of the probabilistic descriptions in quantum theory by recalling the single photon interference experiments conducted by two commonly used optical interferometric systems: the Mazh–Zehnder interferometer (MZI) and Young’s double-slit system (YDS). Though they are both two-path interference, it is shown that the quantum mechanics can provide deterministic predictions for the measurement outcomes in some special cases by adopting the MZI, while only probabilistic predictions can be provided for the measurement outcomes by adopting the YDS. We hope this work can help to understand the probabilistic nature of quantum mechanics.
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.
This chapter introduces the powerful Feynman path integral approach to quantum mechanics. It is intended for readers who are interested in a non-mathematical approach to quantum physics at the middle and high-school level. Einstein's discovery of light quanta played a major role in the birth of quantum mechanics. Einstein made the bold prediction that light comes as a stream of particles called “light quanta”. Photons, like bullets, carry momentum that applies forces to mechanical objects. The method of phasors that, the authors applied to radio waves works equally for visible photons and even for particles such as electrons and molecules. Feynman's method of all possible paths assumes a large number of possible paths. Single-photon counting experiments and laser technology allow the reader to see deeply into the quantum world. The videos of single-particle interference consolidate the idea of photons and their intrinsic randomness.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.