### Schrödinger, Dirac - Quantum Wave Mechanics

The fundamental idea of wave mechanics was the title of Erwin Schrödinger’s Nobel Lecture in 1933. The lecture explains his ideas on wave mechanics in an easily read manner and provides an excellent introduction to this subject45. The Schrödinger equations provide a sound model for multi-electron atoms that agree extremely well with measured results.

The theory specifies the laws of wave motion that the particles of any microscopic system obey. This is done by specifying, for each system, the equation that controls the behaviour of the wave function, and also by specifying the connection between the behaviour of the wave function and the behaviour of the particle. The theory is an extension of the de Broglie postulate and furthermore, there is a close relation between it and Newton's theory of the motion of particles in macroscopic systems.

A comparison can also be made between the Schrödinger theory and Maxwell's theory of electromagnetism because electromagnetic waves behave in a manner which is very analogous to the behaviour of the wave functions of the Schrödinger theory. The Schrödinger equation tells us the form of the wave function ψ (x, t), if we tell it about the force acting on the associated particle (of mass m) by specifying the potential energy corresponding to the force. In other words, the wave function is a solution to the Schrödinger equation for that potential energy

Schrödinger developed his wave equation by using the de Broglie-Einstein postulates where λ=h/p and ɣ=E/h (h=Planck’s constant) which connect the wavelength λ of the wave function with the linear momentum p (=mv) of the associated particle and connect the frequency ɣ of the wave function with the total energy E of the particle with essentially constant p and E. A further postulate of the wave equation relates the total energy E of a particle to its Kinetic Energy KE and Potential Energy V where E = p/2m + V or E = KE + V.

He used the required consistency with these postulates in his search for an argument that is designed to make the quantum mechanical wave equation seem very plausible, but it must be emphasised that this plausibility argument did not constitute a derivation. In the final analysis, the quantum mechanical wave equation was obtained by a postulate, whose justification is not that it has been deduced entirely from information already known experimentally, but that it correctly predicts results which can be verified experimentally.

This description of Schrödinger’s quantum theory is important to the Theory of Physics in 5 Dimensions because the form of the wave equation used fits more comfortably with the expressions of Physics in 5 Dimensions than the expressions of classical physics.

With v4 defined as the velocity vector of a body as viewed by an observer in classical physics, two key aspects of Physics in 5 Dimensions are that:

1. All objects and particles of have a constant mass m (not varying with v4) and a constant energy E = m c and

2. All bodies have three velocity vectors, namely v4, v5 and c (the speed of light), with the scalar relationship c= v4+ v52. It follows46 that all changes of energy can only be an exchange between kinetic energy K = m v4 c and potential energy V = m v5 c where E= K+ V2.

While the Schrödinger Equation E = p/2m + V is non-relativistic, it can be replaced by the equivalent relativistic Physics in 5 Dimensions term K5 where

E = K5 + V = m v4/(1+v5/c) + V; which is also shown47 to be consistent with E= K+ V2. When v4<<c then v5≈c and we return to the Schrödinger Equation where E = K5 + V ≈ p/2m + V.

In 1928 Dirac developed a relativistic theory of quantum mechanics utilising essentially the same postulates as the Schrödinger theory however incorporating a relativistic element. Dirac’s expression included two new variables which give rise to the spin of the electron. Dirac commented in his Nobel Lecture48: The variables also give rise to some unexpected phenomena concerning the motion of the electron. These have been fully worked out by Schrödinger. It is found that an electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small amplitude superimposed on the regular motion that appears to us. As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light. This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory motion is so high and its amplitude so small. But one must believe in this consequence of the theory, since other consequences of the theory which are inseparably bound up with this one, such as the law of scattering of light by an electron, are confirmed by experiment. This comment by Dirac links particles (in this case electrons) moving with the velocity of light c with a quantum theory and supports the hypothesis of Physics in 5 Dimensions that all matter has a path with the velocity of light.

References:

(45) Schrödinger – Nobel Lecture – The fundamental idea of wave mechanics – December 12. 1933 - https://www.nobelprize.org/uploads/2017/07/schrodinger-lecture.pdf

(46) See pages 42-46 - Physics in 5 Dimensions ISBN: 978-3-96014-233-1

(47) See pages 47-48 - Physics in 5 Dimensions ISBN as above

(48) Schrödinger & Dirac - Nobel lecture – Theory of electrons and positrons – 1933 https://www.nobelprize.org/prizes/physics/1933/ceremony-speech/

The Book by Alan ClarkPhysics in 5 Dimensions - is also available as a PDF file to members of ResearchGate here.