Hans de Vries
Nov20-08, 06:30 AM
The Lorentz Force and Maxwell's equations derived from Klein Gordon's equation
.
http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdf
I posted several new chapters of my book lately, mostly involving the Klein Gordon equation.
This chapter shows how the Lorentz Force has a quantum mechanical base and is the
result of the U(1) group represention of the complex Klein Gordon field with electric charge.
( The interaction between A^\mu and \psi goes via the phase of the field )
After some introductions on the Aharonv-Bohm effect the Wilson loops are discussed in
combination with the U(1) symmetry. Thereafter we go to the derivation.
First the acceleration operator is obtained by applying the Hamiltonian twice on the position
operator of the Klein Gordon field.
Applying the acceleration operator on the Klein Gordon field with interaction gives the
the Lorentz Force, the acceleration is a result of the Lorentz Force.
There are lots of illustrations, for instance figures 11.3 through 11.6 show how the wave
nature leads to each of the individual components of the Lorentz force.
Regards, Hans
The Lorentz force derived from Klein Gordon's equation
11.1 Klein Gordon equation with EM interaction. . . . . . . . 2
11.2 Aharonov Bohm effect and experiments . . . . . . . . . . 4
11.3 The scalar phase and Wilson Loops. . . . . . . . . . . . 7
11.4 Lorentz force from the acceleration operator . . . . . . 10
11.5 Parallel electric Lorentz force term . . . . . . . . . . 14
11.6 Orthogonal electric Lorentz force term . . . . . . . . . 15
11.7 Parallel magnetic Lorentz force term . . . . . . . . . . 16
11.8 Orthogonal magnetic Lorentz force term . . . . . . . . . 17
11.9 The total four-vector Lorentz force. . . . . . . . . . . 18
11.10 Maxwell's equations. . . . . . . . . . . . . . . . . . . 20
11.11 Covariant derivative and gauge invariance. . . . . . . . 23
.
http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdf
I posted several new chapters of my book lately, mostly involving the Klein Gordon equation.
This chapter shows how the Lorentz Force has a quantum mechanical base and is the
result of the U(1) group represention of the complex Klein Gordon field with electric charge.
( The interaction between A^\mu and \psi goes via the phase of the field )
After some introductions on the Aharonv-Bohm effect the Wilson loops are discussed in
combination with the U(1) symmetry. Thereafter we go to the derivation.
First the acceleration operator is obtained by applying the Hamiltonian twice on the position
operator of the Klein Gordon field.
Applying the acceleration operator on the Klein Gordon field with interaction gives the
the Lorentz Force, the acceleration is a result of the Lorentz Force.
There are lots of illustrations, for instance figures 11.3 through 11.6 show how the wave
nature leads to each of the individual components of the Lorentz force.
Regards, Hans
The Lorentz force derived from Klein Gordon's equation
11.1 Klein Gordon equation with EM interaction. . . . . . . . 2
11.2 Aharonov Bohm effect and experiments . . . . . . . . . . 4
11.3 The scalar phase and Wilson Loops. . . . . . . . . . . . 7
11.4 Lorentz force from the acceleration operator . . . . . . 10
11.5 Parallel electric Lorentz force term . . . . . . . . . . 14
11.6 Orthogonal electric Lorentz force term . . . . . . . . . 15
11.7 Parallel magnetic Lorentz force term . . . . . . . . . . 16
11.8 Orthogonal magnetic Lorentz force term . . . . . . . . . 17
11.9 The total four-vector Lorentz force. . . . . . . . . . . 18
11.10 Maxwell's equations. . . . . . . . . . . . . . . . . . . 20
11.11 Covariant derivative and gauge invariance. . . . . . . . 23