In this paper we first show that, there exists a precise mechanical analogue for the one-dimensional version of Schrodinger's original 4th-order, real-valued matter-wave equation. It is a composite, flexural-shear beam supported on distributed elastic springs. Nevertheless, in spite of this finding, this paper shows that it is not possible to construct a physically realizable mechanical analogue for Schrodinger's 2nd-order, complex valued matter-wave equation which yields lower eigenvalues; therefore, lower energy levels than these predicted with his original 4th-order, real-valued matter-wave equation.
In this paper we first show that, there exists a precise mechanical analogue for the one-dimensional version of Schrodinger's original 4th-order, real-valued matter-wave equation. It is a composite, flexural-shear beam supported on distributed elastic springs. Nevertheless, in spite of this finding, this paper shows that it is not possible to construct a physically realizable mechanical analogue for Schrodinger's 2nd-order, complex valued matter-wave equation which yields lower eigenvalues; therefore, lower energy levels than these predicted with his original 4th-order, real-valued matter-wave equation.
Using a variational formulation, we show that Schrodinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential V(r), produces the precise eigenvalues of Schrodinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative (repelling) energy levels. Schrodinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential V(r), at the expense of missing the negative (repelling) energy levels.
Using a variational formulation, we show that Schrödinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential V ( r ) , produces the precise eigenvalues of Schrödinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. The variational forms of the matter-wave equations are computed numerically with a Ritz-spline method and we show how this method handles accurately discontinuous potentials with singular derivatives. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative, mirror energy levels. Schrödinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential V ( r ) , at the expense of missing the negative, mirror energy levels. • Variational formulations are derived for Schrödinger’s 2nd- and 4th-order matter-wave equations. • A novel Ritz spline method is developed to solve numerically for the eigenenergies and corresponding wave functions. • Schrödinger’s 4th-order matter-wave equation is real-valued, but includes spatial derivatives of the potential. • The 4th-order, real-valued formulation produces all eigenvalues of the 2nd-order equation, plus a set of mirror eigenvalues. • A real-valued description of non-relativistic quantum mechanics exists, based on Schrödinger’s 4th-order equation.
vol. 81, 1926 paper, Schrödinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is fourth-order in space and second-order in time. In the view of the mathematical difficulties associated with the eigenvalue analysis of a fourth-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrödinger splits the fourth-order real operator into the product of two, second-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic second-order, complex-valued wave equation. In this paper, we show that Schrödinger's original fourth-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his second-order, complex-valued wave equation that predicts with remarkable accuracy the energy levels observed in the atomic line spectra of the chemical elements. Accordingly, the fourth-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that quantum mechanics can only be described with the less stiff, second-order, complex-valued wave equation.
The wave function Ψ satisfies the Schrödinger equation for a free particle. Formally, the Schrödinger equation generates the magnitude of mechanical motion of the zero order 0 p = mv 0 (in the sense that it is contained in the Schrödinger equation. Comparison of the wave function Ψ and its gradient implies a formal analogue of the Schrödinger equation, which generates the magnitude of mechanical motion of the first order 1 p = mv 1. From the comparison of the wave function and its time derivative yields a formal analogue of the Schrödinger equation, which generates the magnitude of mechanical motion of the second order 2 p = mv 2/2!. The values of mechanical motion of the zero, first, and second orders are known.Obviously, other formal analogs of the Schrödinger equation can generate mechanical motion of other orders.The aim of the work is to establish such quantities and related regularities that may be of interest, which makes the study relevant. hovering. Then the spatial derivatives will be one-dimensional. The magnitude of the mechanical movement of the third order is 3 p = mv 3/3!. This value is Umov's integral vector for kinetic energy. The magnitude of the mechanical movement minus the first order -1 p = mv -1 is a reverse impulse. The meaning of this quantity and its relevance is established by the theorem: in a hy- drogen-like atom, the quantity m v-1 is quantized. A fixed (unchanged) quantum is a quantity m v-1 correspond- e e 0 ing to the basic energy level. Almost all of the results obtained were a consequence of the use of quantum mechanical differential equations, however, the results themselves are predominantly macromechanical. The quantities of mechanical motion of various orders are generated by formal analogs of the Schrödinger equation. In all formal analogs of the Schrödinger equation, the orders of the partial derivatives differ by one. For quantities of motion with a positive degree of velocity, the order of the temporal derivatives is higher than that of the spatial ones. For quantities with a negative degree, the order of spatial derivatives is higher.
th , 2012; revised: Jan. 8 th , 2013; accepted: Jan. 16 th , 2013 Abstract: In order that students have a whole realization of the wave mechanics foundation, we review the two impor- tant enlightened steps in details, i.e. the analogues between mechanics and optics, de Broglie matter wave. Based on the duality of wave and particle, we report the whole process that Schrodinger founded his wave mechanics. We introduce Schrodinger's interesting and deep method of deriving wave equation to quantum mechanics teaching.
Schrödinger’s wave mechanics, developed in a series of papers published one century ago, offered an exceptionally intuitive route into the new atomic physics. The later axiomatic formulation of quantum mechanics brought indispensable mathematical precision, but it can also conceal the physical images that guided the original discovery. We revisit the optical–mechanical analogy behind Schrödinger’s construction and use it as a visual guide to stationary states, superposition, field modes, and gauge interactions. The analogies are deliberately heuristic and their limitations are stated: a quantum wavefunction is not, in general, a classical wave in ordinary space, and discrete classical modes do not by themselves imply quantized energy. Nevertheless, the wave viewpoint makes it possible to see the operator language of first and second quantization as the compact algebraic expression of an underlying mode structure. The purpose of this article is therefore not to replace the modern formalism, but to recover the physical intuition that makes that formalism intelligible.