Abstract
We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations.
Highlights
In quantum theory, a classical observable, which is modeled by a real scalar variable, is replaced by a quantum operator, which is analogous to an infinite-dimensional matrix operator
It is quite easy to show that the Heisenberg equations imply the quantum Hamilton equations, namely that: d pi
We have shown that quantum Hamilton equations can be derived from the Heisenberg equations with the fundamental commutator as given in (1)
Summary
A classical observable, which is modeled by a real scalar variable, is replaced by a quantum operator, which is analogous to an infinite-dimensional matrix operator. It can be shown that quantum Hamilton equations can be derived from the Heisenberg equations of motion after invoking the fundamental commutator [3]. There is usually a clear analogy and a mathematical homomorphism between the classical and quantum theories of many systems, such as electromagnetic fields This streamlines the derivation of many quantum equations of motion. It explains why the equations of motion for quantum electromagnetics resemble those of classical electromagnetics [5] Using this mathematical homomorphism, one can derive the quantum Maxwell equations without using mode decomposition (or Fourier decomposition), a procedure followed in most textbooks [6,7,8]. The quantum Maxwell equations can be derived [5]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
