Seeking the unknown dynamics obeyed by a particle gives rise to the de Broglie wave representation, without the need for physical assumptions specific to quantum mechanics. The only required physical assumption is conservation of momentum μ. The particle, of mass m, moves through free space from an unknown source-plane position a to an unknown coordinate x in an aperture plane of unknown probability density pX(x), and then to an output plane of observed position y=a+z. There is no prior knowledge of the probability laws p(a,M),p(a) or p(M), with M the particle momentum at the source. It is desired to (i)optimally estimate a, in the sense of a maximum likelihood (ML) estimate. The estimate is further optimized, by minimizing its error through (ii) maximizing the Fisher information about a that is received at y. Forming the ML estimate requires (iii) estimation of the likelihood law pZ(z), which (iv) must obey positivity. The relation pZ(z)=|u(z)|*2≥0 satisfies this. The same u(z) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u(z) achieves aims (i)–(iv). The output is parametrized by a free parameter K. For a choice K=0, the result is u(z)=δ(z), indicating classical motion. Or, for a finite, empirical choice K=h¯(Planck’s constant), u(z) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P(μ) on momentum μ. For a definite momentum μ, u(z) becomes a sinusoid of wavelength λ= h¯/μ, the de Broglie result.
Seeking the unknown dynamics obeyed by a particle gives rise to the de Broglie wave representation, without the need for physical assumptions specific to quantum mechanics. The only required physical assumption is conservation of momentum μ. The particle, of mass m, moves through free space from an unknown source-plane position a to an unknown coordinate x in an aperture plane of unknown probability density pX(x), and then to an output plane of observed position y=a+z. There is no prior knowledge of the probability laws p(a,M),p(a) or p(M), with M the particle momentum at the source. It is desired to (i)optimally estimate a, in the sense of a maximum likelihood (ML) estimate. The estimate is further optimized, by minimizing its error through (ii) maximizing the Fisher information about a that is received at y. Forming the ML estimate requires (iii) estimation of the likelihood law pZ(z), which (iv) must obey positivity. The relation pZ(z)=|u(z)|*2≥0 satisfies this. The same u(z) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u(z) achieves aims (i)–(iv). The output is parametrized by a free parameter K. For a choice K=0, the result is u(z)=δ(z), indicating classical motion. Or, for a finite, empirical choice K=h¯(Planck’s constant), u(z) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P(μ) on momentum μ. For a definite momentum μ, u(z) becomes a sinusoid of wavelength λ= h¯/μ, the de Broglie result.
In (1), a variational approach is applied to Fisher’s information (with constraints) to obtain a differential equation in P(x)=probability to be at x. It is suggested the equation may be simplified using P(x)=W(x)W(x) where W(x) is real. The result is the nonrelativistic Schrodinger equation with mass=1. As a first observation, we note that Fisher’s information, which is a purely statistical quantity, should not necessarily yield a nonrelativistic equation which is an approximation of relativistic results. In particular, we argue that Fisher’s information is used to find <pp> not necessarily 1/2m <pp>, but this is a minor point. A more important issue, we argue, is the meaning of P(x) in a bound system. A classical bound system is sometimes said to have a P(x)=C/v(x) i.e. P(x)dx is proportional to dt, the amount of time a particle spends in each equally sized dx. In the high energy limit, the envelope of a quantum bound probability P(x)=W(x)W(x) (W(x)=wavefunction) is said to roughly match C/v(x), yet this density does not follow from the extremization of Fisher’s information (together with two general constraints which yield E and V(x)=potential). This begs the question: What is P(x)? If one considers a particle moving at constant speed (i.e constant momentum) then P(x)=constant because each dx point carries the same probability. Classically a particle placed in a box with infinite potential walls would have a P(x) =constant and p and -p motion would be separated in time. Thus if P(x) is to have meaning in a box, one must conclude that the probabilities associated with p and -p are not constant, but rather functions of x. This then leads to the notion of a complex new kind of probability (which maps to P(x)=constant) for a free particle i.e. W(x) =cos(px) + i sin(px) so that the modulus W*(x)W(x)=P(x)= 1. These ideas seem to go beyond the notion of Fisher information. What then is the role of Fisher information? We argue it is related to two very specific moments of the distribution cos(px) or rather W(x) = Sum over p a(p)cos(px) with a(p)=a(-p), namely and <pp>. In classical physics there exists p and pp at each x point (but one is the square of the other), but with distributions <p does not equal to <pp> in general. Both and <pp> are linked to physical concepts which exist in Newtonian mechanics, namely impulse and work. With the presence of cos(px) one has a combination of Newtonian mechanics and a statistical distribution. We argue that the role of Fisher’s information is to allow one to transform from to <pp>. In particular, -idW/dx appears in Fisher’s information (and is linked to ), but the specific role of a variation is to transform it into <pp> which appears in a conservation of energy equation. We argue that this is why Fisher’s information may be used to derive the Schrodinger equation.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
A formulation is proposed for the behavior of a particle in a box based on two assumptions: (1) particles have an associated phase which changes periodically with time as defined by de Broglie, and (2) the transmission of phase information follows optical principles, including reflection at potential boundaries and relativistic Doppler effects. This approach defines optimal trajectories by matching the phase of a source and its reflections. The momentum of the particle is guided by a de Broglie wave which follows the equation R exp {i[w0 (1-v^2/c^2)^-1/2][(t - vz)/c^2]} cos {[w0(1-v^2/c^2)^-1/2/c](vt-z)} This wave is generated by reflections occurring an even number of times, the phase and amplitude of which correspond to Lorentz transforms relative to the time and position, respectively. Nodes are produced by reflections occurring an odd number of times, which can be conceived to generate a wave traveling in the opposite direction and shifted by a phase constant π. Application to an individual particle in an infinite potential well affords the same results for energies and nodes as derived by the Schrödinger equation.
It is shown that the assumption that nature acts to make optimum estimates of position maximally in error leads to the Schrödinger (energy) wave equation (SWE). In this way, the SWE follows from a simple statement of uncertainty. The approach grows out of probability estimation theory, in particular the principle of minimum Fisher information (or maximum Cramer–Rao bound). The minimized Fisher information turns out to be proportional to the mean kinetic energy of the particle. This approach is an attractive supplement to conventional ways of introducing quantum mechanics to students, since it avoids the use of imperfect physical models (such as vibrating strings) or the immediate need for complex momentum operators, and follows from a plausible assumption as to how nature works.
The phase corresponding to a certain path in the Feynman path integral is derived with the help of the de Broglie hypothesis. Zusammenfassung. Die einem bestimmten Pfad entsprechende Phase im Feynman-Pfadintegral wird mit Hilfe der de Broglie-Hypothese abgeleitet. In the beginning quantum mechanics was built up by analogy with the formulation of classical mechanics in terms of the Hamilton function. In 1933, Dirac [2] announced the idea of formulating quantum mechanics on the basis of the Lagrange function. He started from the analogy between the treatment of classical motion as a canonical transformation and the transformational properties of operators with respect to a time translation. This idea of Dirac was developed further by Feynman [3], who regarded a quantum-mechanical motion from one space-time point to another as a superposition of motions along all possible paths connecting the two points. Each path contributes to the total probability amplitude a term proportional to where is the classical action corresponding to this path and is Planck's constant. Feynman [3] adopted the form of the phase in the expression (1) as a postulate. Another way, however, of deriving the phase is to use the de Broglie hypothesis [1], as shown below. According to this hypothesis, it is possible to associate any particle with a certain frequency proportional to its energy (Hamiltonian function), i.e., and a wave vector proportional to the particle's momentum as Now consider, in the sense of Feynman's formulation, a particle moving along a certain path from point 1 to point 2. During the motion, let and be the radius vectors of the particle at the times t, t+ dt respectively. According to the de Broglie hypothesis, we can associate a plane wave with the particle in the neighbourhood of these points. The phase change of this wave along the particle's path during the motion between the points and consists of space and time phase changes: The term in the parentheses is precisely the value of the Lagrange function at the point . The phase change of the wavefunction corresponding to the whole path can be obtained by integration of the term expressing the infinitesimal phase change: In this way, on the basis of the de Broglie hypothesis, we have reached full agreement with Feynman's postulate: the phase of the probability amplitude corresponding to a certain path is equal to , where is the classical action corresponding to this path. References [1] de Broglie L 1923 Comptes rendus 177 548 - 550 [2] Dirac P A M 1933 Phys. Z. der Sowjetunion 3 64 [3] Feynman R 1948 Rev. Mod. Phys. 2 367
This paper derives the de Broglie matter wave and the Born rule as logical consequences of two fundamental principles: (i) the mathematical necessity of Fourier duality between position and wave number, combined with the physical requirement of continuous spatial translations (enforced by translation symmetry), and (ii) Noether’s theorem applied to Lagrangians with no explicit spatial dependence. Translation invariance of the laws of physics, assuming a uniform background, then identifies linear momentum p as the conserved Noether charge conjugate to position x. This yields the de Broglie relation p ∝ k through equivalence of the Noether (symmetry) and Fourier (duality) pictures of conjugate variables. The resulting de Broglie wave is interpreted as a virtual, unobservable predisposition field: an alternative Fourier-space description of the same particle. By associating wave intensity with statistical energy flux (for single particles emitted sequentially or coherent ensembles), we demonstrate that the probability density of detection must be proportional to the squared modulus of the wave function, P(x) ∝ |ψ(x)|². This yields the Born rule without invoking interference experiments or photon-specific classical analogies for non-photonic matter waves. The derivation relies solely on symmetry principles, Fourier mathematics, and the indivisibility of quantum detection events, offering a reconstruction free of ad-hoc postulates.