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arXiv:2503.20077 · New Journal of Physics

Enhancing wave–particle duality

Arwa Bukhari, Daniel Hodgson, Sara Kanzi, Robert Purdy, Almut Beige

Published Jul 28, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract To enhance the consistency between the quantum descriptions of waves and particles, we quantise mechanical point particles in this paper in the same physically-motivated way as we previously quantised light in quantum electrodynamics (Bennett et al 2016 Eur. J. Phys. 37 014001). To identify the relevant Hilbert space, we notice that mechanical particles can occupy any position x while moving at any velocity v. Afterwards, we promote the classical states (x, v) to pairwise orthogonal quantum states | x , v ⟩ and demand that these evolve according to Newton’s equations of motion. The resulting quantum theory is mass-independent, when Newton’s equations of motion are mass-independent, as one would expect. The basic formulation of quantum mechanics emerges from quantum mechanics in configuration space as a semi-classical approximation when a fixed mass is imposed and several other adjustments are made.

DOI
10.1088/1367-2630/adf506

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Abstract To enhance the consistency between the quantum descriptions of waves and particles, we quantise mechanical point particles in this paper in the same physically-motivated way as we previously quantised light in quantum electrodynamics (Bennett et al 2016 Eur. J. Phys. 37 014001). To identify the relevant Hilbert space, we notice that mechanical particles can occupy any position x while moving at any velocity v. Afterwards, we promote the classical states (x, v) to pairwise orthogonal quantum states | x , v ⟩ and demand that these evolve according to Newton’s equations of motion. The resulting quantum theory is mass-independent, when Newton’s equations of motion are mass-independent, as one would expect. The basic formulation of quantum mechanics emerges from quantum mechanics in configuration space as a semi-classical approximation when a fixed mass is imposed and several other adjustments are made.
To enhance the way in which wave-particle duality is implemented in the modelling of quantum mechanical systems, Bukhari et al. [New J. Phys. 27, 084501 (2025)] recently introduced an alternative approach to quantum mechanics, namely quantum mechanics in configuration space. This formalism is based on a physically motivated quantisation of Newtonian mechanics and promotes the classical position-velocity states (x,v) to pairwise distinguishable quantum states. The resulting |x,v> states form the basis of the Hilbert space of individual quantum mechanical particles and evolve along classical trajectories. In this paper, we consider the modelling of a mechanical particle in free space and put quantum mechanics in configuration space into context. It is shown that this formalism increases the continuity between quantum and classical mechanics by avoiding a conceptual inconsistency associated with the definition of momentum in canonical quantisation. In addition, we emphasise that standard quantum mechanics and quantum mechanics in configuration space are based on two distinct formulations of classical mechanics.
Quantum mechanics is one of basics theories of modern physics, and it is also a theory established with some debates of assumptions. However, in the teaching process, if the assumption of wave-particle duality is directly given, it will be particularly difficult for students to understand. Therefore, based on the teaching practice and previous research, we assume that light corresponds to a kind of real particle starting from the classical electromagnetic theory, according to the description methods of wave optics and geometric optics, and the form of functional equations. Subsequently, employing Planck's quantum conditions, we discuss the connection between the wave nature and particle nature of light. And then the Planck-Einstein relation and the de Broglie relationship are derived, which reflect the wave-particle duality of light and real particle.
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
This is the third of three chapters that apply the concepts of Chapters 1, 2, and 3 to the three principal types of waves: mechanical, electromagnetic, and quantum. Although mechanical waves may be the most obvious in everyday life, and electromagnetic waves may be the most useful to our technological society, a case can be made that quantum waves are the most fundamental. As you'll see in this chapter, every bit of matter in the Universe behaves like a wave under certain circumstances, so it's hard to imagine anything more fundamental. If you're thinking that life in the macroscopic world moves along just fine without considering quantum effects, you should realize that the laws of classical physics are approximations, and all the quantum weirdness described in this chapter blends smoothly with classical physics when they overlap. Much of that weirdness comes from the dual wave and particle nature of matter and energy, and one of the goals of this chapter is to help you understand what's actually doing all this waving. This chapter begins with a comparison of the characteristics of waves and particles in Section 6.1, followed by a discussion of wave–particle duality in Section 6.2. You can read about the Schrödinger equation and probability wavefunctions in Sections 6.3 and 6.4, and quantum wave packets are discussed in Section 6.5. Wave and particle characteristics Before studying modern physics, most students think of particles and waves as belonging to fundamentally different categories of objects. That's understandable, because particles and waves have several very different characteristics, including the way in which they occupy space, the way they travel through openings, and the way they interact with other particles or waves. Here's a summary of some of the differentiating characteristics between particles and waves. Occupying space. Particles exist in a well-defined amount of space; if you could pause time at some instant during a tennis match, all of the spectators would agree on the position of the ball at that instant.
Dispersion dynamics applies wave-particle duality, together with Maxwell’s electromagnetism, and with quantization E = hν = ħω (symbol definitions in footnote) and p = h/λ = ħk, to special relativity E2 = p2c2 + m2c4. Calculations on a wave-packet, that is symmetric about the normal distribution, are partly conservative and partly responsive. The complex electron wave function is chiefly modelled on the real wave function of an electromagnetic photon; while the former concept of a “point particle” is downgraded to mathematical abstraction. The computations yield conclusions for phase and group velocities, vp⋅vg = c2 with vp ≥ c because vg ≤ c, as in relativity. The condition on the phase velocity is most noticeable when p≪mc. Further consequences in dispersion dynamics are: derivations for ν and λ that are consistently established by one hundred years of experience in electron microscopy and particle accelerators. Values for vp = νλ = ω/k are therefore systematically verified by the products of known multiplicands or divisions by known divisors, even if vp is not independently measured. These consequences are significant in reduction of the wave-packet by resonant response during interactions between photons and electrons, for example, or between particles and particles. Thus the logic of mathematical quantum mechanics is distinguished from experiential physics that is continuous in time, and consistent with uncertainty principles. [Footnote: symbol E = energy; h = Planck’s constant; ν = frequency; ω = angular momentum; p = momentum; λ = wavelength; k = wave vector; c = speed of light; m = particle rest mass; vp = phase velocity; vg = group velocity].