We derive the Schrödinger equation, starting from principles other than that of superposition principle and of correspondance between physical quantities and linear operators as commonly used. We assume that the particle is carried by a compressible flow and that its density of momentum flux is only dependant on the spatial derivatives of the density . Then the condition that the flow can be derived from a potential is sufficient to show that satisfies the Schrödinger equation. This implies that the superposition principle is a property of some potential compressible fluids.
We derive the Schrödinger equation, starting from principles other than that of superposition principle and of correspondance between physical quantities and linear operators as commonly used. We assume that the particle is carried by a compressible flow and that its density of momentum flux is only dependant on the spatial derivatives of the density . Then the condition that the flow can be derived from a potential is sufficient to show that satisfies the Schrödinger equation. This implies that the superposition principle is a property of some potential compressible fluids.
Abstract The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
In (1), the Schrodinger equation is derived using a hydrodynamical continuity of density approach. Thus, spatial density represents x probability. There, however, is also a flow velocity field in the problem v(x) and it is assumed this is irrotational. From this it is stated in (1) that v(x) is proportional to 1/m d/dx S(x) ((1)) where S(x) is some function and m is mass. In this note, we argue that a particle moving in one dimension with constant p (momentum) is automatically irrotational, but it should satisfy ((1)) from (1). In other words, a constant p particle hence constant velocity is associated with a function of x, S(x) even though its density in space is not. If one considers S(x)=ln(B(x)), then ((1)) is equivalent to: p = d/dx ln(B(x)). If one tries to associate B(x) with a probability, then ln(B(x)) is information = px. B(x), however, is a different kind of x probability from density(x) which should be constant for a free moving particle. In fact, using -id/dx instead of d/dx and exp(ipx) for B(x), one finds exp(ipx) as a second x probability. Its modulus yields the constant value of density(x). Thus the irrotational flow assumption leads to p= dS/dx. Incidentally, this form is equivalent to Fermat’s least time principle for light with p=p (x component). We argue that applying p=dS/dx associates an x probability distribution even to a constant p, so the assumption p=dS/dx is essentially a quantum mechanical statement, even if one does not derive the Schrodinger equation. The time independent Schrodinger equation may be obtained by using exp(ipx) as a relative probability and using energy conservation.
The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .
PACS number: 03.75.-b Abstract. Electromagnetic waves and fluids have locally conserved mechanical properties associated with them and we may expect these to exist for matter waves. We present a semiclassical description of the continuity equations relating to these conserved properties of matter-waves and derive a general expression for their respective fluxes. It has long been known that electromagnetic waves have associated mechanical properties. Maxwell described the radiation pressure exerted by light [1] and Poynting identified the momentum carried by light [2] and that angular momentum was associated with circular polarisation [3]. The energy, momentum and angular momentum of electromagnetic waves are particularly important as these quantities are all conserved. Associated with the density of each of these quantities is a flux density (or current density) which is related to the density via continuity equations [4]. The creation of the first Bose-Einstein condensates (BECs) in dilute atomic gases in 1995 [5, 6, 7] brought about the possibility of creating coherent matter waves. A BEC in a trap is analogous to an electromagnetic field in a laser cavity, in that macroscopic occupation of a single quantum state occurs. Just as a laser cavity is a source of coherent light, a BEC can be a source of coherent matter waves. To continue this analogy we assert that matter waves will have associated mechanical properties and present a general theory of the local mechanical properties of matter waves. We give, as examples, the densities and flux densities of the matter wave particle number and energy. The same theory applies to their momentum and angular momentum. The local conservation of particles is a well established concept in fluid mechanics. It states that in any volume element of a fluid, no matter how small, the rate of change of particles in an element of fluid is equal to the rate at which the particles flow into the surface of the element. This principle is made quantitative in the following equation [8];