Quantum Spin, Constraints, Degeneracy and Matrices
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2021
|
| Subjects: | |
| Online Access: |
https://zenodo.org/record/5274110 |
Daftar Isi:
- In classical physics, angular momentum is defined by rxp (r, p vectors) which is constrained motion because it is angular momentum. There is also the idea of degeneracy present because for a fixed angular momentum, say a classical object orbiting another in a circle, the magnitude of velocity is the same at each point. In classical electrodynamics, one may consider the case of a photon, i.e. Maxwell’s equations without source or current terms. These lead to El (electric) and B magnetic fields existing in a plane perpendicular to the photon motion (and to each other). In addition, one may take the four equations and obtain a wave equation for both El and B. Given a wave equation, one has a degeneracy in the direction of El. Alternatively, one may create a circularly polarized El which has a constant magnitude, but moves in a circle like angular momentum which it is. In fact, this circularly polarized motion represents spin 1, but is seen as classical motion. To make a link with quantum mechanics, one may take the two classical Maxwell’s equations: dEl/dt = grad x B and dB/dt = -grad x E and write them in terms of a matrix equation (with the matrix coming from i times the Levi-Civita symbol) i.e. id/dt (El + iB) = (S dot p) (El + iB) ((1)) where p=-i grad and Sn= i enjk and enjk is the Levi -Civita symbol. The consequence of using a matrix is that it has eigenfunctions which may be related to a constraint i.e. there may be a fixed “angular momentum” i.e. spin and one finds projection eigenfunctions. This occurs for ((1)) which is a matrix equation (containing spin) created from purely classical equations. The eigenvalues of the particular matrix are discrete (i.e. there is quantization) and one may form a linear combination to find the most general math solution which is the opposite of what one would do in classical physics. For example, for a spin one object, an m projection of 1 means angular motion in one direction while -1 is in the opposite direction. In classical physics, where one can track a particle at each point at each time, one physically distinguishes between clockwise and counterclockwise motion. In quantum mechanics, one does not distinguish states at a point, but globally and so both clockwise and counterclockwise states may exist in a wavefunction, but it may collapse to one or the other when a measurement occurs. We examine the behaviour of matrices, constraints and degeneracies in the photon wavefunction created from Maxwell’s classical electromagnetism equations, because we wish to extend these ideas to the Dirac electron. In general, physics texts speak of linearizing the Klein-Gordon wave equation. What happens during this linearization is that matrices appear which follow constraint equations. There have eigenfunctions and eigenvalues which we argue is the pattern for angular motion i.e. spin. The type of matrix which appears governs the spin present and again one adds eigenfunctions for the most general solution in quantum mechanics. This is completely opposite of classical physics, but explains the Stern-Gerlach experiment. Thus, we argue that matrices which appear (constrained) when one linearizes a quadratic differential equation linking energy and momentum in quantum mechanics points to degenerate (because it does not appear in the p, E equation explicitly), constrained motion i.e. angular motion or spin.