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arXiv:2601.12176

Wave Phenomena and Wave Equations

Gerd Leuchs, Mojdeh S. Najafabadi

physics.flu-dynquant-phSubmitted Jan 17, 2026• Announced Jan 20, 2026• 1 citation
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Abstract

For any kind of wave phenomenon one can find ways to derive the respective dispersion relation from experimental observations and measurements. This dispersion relation determines the structure of the wave equation and thus characterizes the dynamics of the respective wave. Different wave phenomena are thus governed by different differential equations. Here we want to emphasize the experimental approach to matter waves, but before doing so we will discuss and test the procedure for other types of waves, in particular water waves.

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For any kind of wave phenomenon one can find ways to derive the respective dispersion relation from experimental observations and measurements. This dispersion relation determines the structure of the wave equation and thus characterizes the dynamics of the respective wave. Different wave phenomena are thus governed by different differential equations. Here we want to emphasize the experimental approach to matter waves, but before doing so we will discuss and test the procedure for other types of waves, in particular water waves.
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In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
Taking into account wave-particle duality and the uncertainty principle, a nonlinear Schrödinger wave equation (NLSE) for the description of dissipative systems can be constructed by applying only Newton's form of classical mechanics. A comparison between our NLSE for dissipative systems and the linear time-dependent SE for conservative systems shows that, for free motion and motion in a harmonic potential, both equations have Gaussian wave-packet solutions. The time dependence of the maximum of these functions is governed by the classical (real) Newtonian equation of motion. While the time dependence of the width of the wave packet can be determined from a nonlinear complex Bernoulli/Ricatti type differential equation. This can be done by transformation into a Newton-type equation for a real quantity α (which is proportional to the square root of the width), including a kind of “centrifugal force” term. This equation can be solved analytically and when taken with Newton's equation of motion for the mean value of the position, it forms a so-called Ermakov system. For the linear SE equation, this implies the existence of a dynamical invariant. For our NLSE we also obtain such an exact invariant, but now for a dissipative system. Furthermore, this complex nonlinear equation can be linearised using the logarithmic derivative of a complex quantity λ, thus yielding a complex Newtonian equation of motion. The absolute value of λ is equivalent to α, the phase angle can be determined using a relation that connects real and imaginary parts of λ. This relation is linked to a kind of “conservation of angular momentum” for the quantity λ in a complex plane. The relations between the above-mentioned quantities will be discussed in detail.
The concept of additional spatial dimensions has long been invoked to explain phenomena in high‑energy physics and cosmology. In this work we propose a fundamentally different paradigm: the presence of a hidden temporal coordinate within the response functions of quantum materials. We argue that the intrinsic dynamics of electrons, phonons, and collective excitations are governed not only by the observable laboratory time t but also by an auxiliary time parameter a_h that is not directly accessible to conventional measurement. This hidden time coordinate manifests itself in the structure of the Green's functions and in the evolution of many‑body states, leading to observable consequences such as anomalous dispersion, non‑Markovian relaxation, and time‑crystalline behaviour. We develop a formalism that incorporates the hidden time into the standard path‑integral and operator approaches, derive modified equations of motion, and discuss potential experimental signatures in ultrafast spectroscopy. The framework suggests that quantum materials can serve as a laboratory for exploring higher‑dimensional temporal physics, opening pathways for discovering new time‑related phases and for achieving unprecedented control over quantum dynamics.