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CRC Press eBooks

Quantum Principles and the Probability Amplitude

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

Twelve mathematical statements that define the quantum mechanics of interference and entanglement are listed in the chapter. The importance of superposition probability amplitudes, via the Dirac–Feynman principle, in quantum mechanics is highlighted. Particular attention is given to the proper description of experimental configurations via probability amplitudes. The refinement of spatial accuracy, only limited by Nature’s availability of shorter and shorter wavelengths, in N -slit quantum interferometers is described. The transition from superposition probability amplitudes to quantum probabilities is discussed. In all this, the essential importance of the nonlocality of the photon is emphasized. The importance of the probability amplitude of quantum entanglement at the foundations of quantum physics is discussed relative to quantum interference. The intrinsic link between these two principles is highlighted via a series of transparent mathematical statements.

DOI
10.1201/9781003398707-22

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Twelve mathematical statements that define the quantum mechanics of interference and entanglement are listed in the chapter. The importance of superposition probability amplitudes, via the Dirac–Feynman principle, in quantum mechanics is highlighted. Particular attention is given to the proper description of experimental configurations via probability amplitudes. The refinement of spatial accuracy, only limited by Nature’s availability of shorter and shorter wavelengths, in N -slit quantum interferometers is described. The transition from superposition probability amplitudes to quantum probabilities is discussed. In all this, the essential importance of the nonlocality of the photon is emphasized. The importance of the probability amplitude of quantum entanglement at the foundations of quantum physics is discussed relative to quantum interference. The intrinsic link between these two principles is highlighted via a series of transparent mathematical statements.
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The superposition principle is a founding principle of quantum mechanics; we have already made use of it in interpreting the Young's slit experiment. Quantum mechanics is a linear theory, and so it is natural that vector spaces play an important role in it. We shall see that a physical state is represented mathematically by a vector in a space whose characteristics we shall define; this is called the space of states . A second founding principle, which can also be deduced from the Young's slit experiment, is the existence of probability amplitudes. These probability amplitudes will be represented mathematically by scalar products defined on the space of states. In the theory of waves, the use of complex numbers is just a convenience, but in quantum mechanics the probability amplitudes are fundamentally complex numbers – the scalar product will a priori be a complex number. Physical properties like momentum, position, energy, and so on will be represented by operators acting in the space of states. In this chapter we shall introduce the essential properties of Hilbert spaces, that is, vector spaces on which a positive-definite scalar product is defined, and we shall limit ourselves to the case of finite dimension. This restriction will be lifted later on, because the space of states is in general of infinite dimension. The mathematical theory of Hilbert spaces of infinite dimension is much more complicated than that of spaces of finite dimension, and we shall put off studying them until Chapter 7. The reader familiar with vector spaces of finite dimension and operators in such spaces can proceed directly to Chapter 3 after reviewing the notation.
The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.
The experimental origin of quantum mechanics is highlighted, and essential features of the photon are described. The philosophical path to quantum entanglement, as defined by the works of EPR–Bohm and Aharonov–Bell, and the physics path to quantum entanglement, as defined by the works of Dirac–Wheeler–Pryce and Ward, are introduced and discussed. The probability amplitude for quantum Entanglement, discovered in 1947, is then introduced prior to a high-level overview of the field. This introduction then concludes with a chapter-by- chapter description of the book and its purpose.