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CRC Press eBooks

Probability Amplitudes

F. J. Duarte

Published Apr 10, 2026• 0 citations
Publisher

Abstract

Quantum mechanics runs on probability amplitudes. Probability amplitudes must accurately describe the physics prior to calculating the quantum probability. Symbolical probability amplitudes are mathematically represented by complex wave functions. Many physicists in the field don’t even mention the words probability amplitude and prefer to refer to wave function, or state, instead. When they do that they miss out on the subtleties of the physics at hand. In this brief chapter we emphasize probability amplitudes à la Dirac since that is the way Feynman and Ward preferred to describe quantum mechanics. The Dirac notation is also ideally suited to describe quantum interference and quantum entanglement.

DOI
10.1201/9781003649434-4

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Quantum mechanics runs on probability amplitudes. Probability amplitudes must accurately describe the physics prior to calculating the quantum probability. Symbolical probability amplitudes are mathematically represented by complex wave functions. Many physicists in the field don’t even mention the words probability amplitude and prefer to refer to wave function, or state, instead. When they do that they miss out on the subtleties of the physics at hand. In this brief chapter we emphasize probability amplitudes à la Dirac since that is the way Feynman and Ward preferred to describe quantum mechanics. The Dirac notation is also ideally suited to describe quantum interference and quantum entanglement.
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.
At this point one might have noticed that both the wave function ^(r, t) and the expansion coefficients have been referred to as probability amplitudes. In this chapter, we introduce the Dirac notation, a way of writing vectors which reveals that y/(r, t) and C*(/) are both expansion coefficients of the state vector | y/{t) >, a vector in an abstract space. The absolute squares of the expansion coefficients are the probabilities of finding the state vector in corresponding eigenstates. The Dirac notation also provides a compact, versa­ tile means for considering any (and all) representatives (energy, position, etc.) o f a state vector.
Amplitudes play a key role in quantum mechanics, as they are the link between theory and experiment. Theorists use quantum mechanics to calculate the amplitudes for various situations, and then experimenters set up appropriate apparatus and take measurements to check the theoretical conclusions. The idea that states can be expanded over a suitable basis is a very important aspect of quantum theory. This is how, in part, quantum theory reflects the contextual nature of quantum reality, that is, that the behaviour of a system is influenced by the experimental context. Every quantum state can be expanded in various ways, depending on what sort of experiment is involved. The first person to introduce symbols such as |φ〉 for quantum states was the celebrated English physicist, Paul Dirac. Dirac notation has become the standard alphabet for working in quantum theory.