In this paper we revise the main aspects of the Hamiltonian analogy: the fact that optical paths are completely analogous to mechanical trajectories. We follow Schr\"{o}dinger's original idea and go beyond this analogy by changing over from the Hamilton's principal function $S$ to the wave function $\Psi$. We thus travel from classical to quantum mechanics through optics.
Deep Dive into From Classical to Quantum Mechanics through Optics.
In this paper we revise the main aspects of the Hamiltonian analogy: the fact that optical paths are completely analogous to mechanical trajectories. We follow Schr"{o}dinger’s original idea and go beyond this analogy by changing over from the Hamilton’s principal function $S$ to the wave function $\Psi$. We thus travel from classical to quantum mechanics through optics.
According to Arnold Sommerfeld mechanics is "the backbone of mathematical physics"; it is a highly elaborated theory emerging from Newton's laws which, in turn, summarize in precise form the whole body of experience.
Classical mechanics describes with great precision the motion of macroscopic bodies, including astronomical bodies, except when velocities and masses involved are excessively large. In the domain of microscopic bodies: atoms, molecules, electrons, etc., the situation is quite different, for the laws of mechanics, as well as those of electromagnetism, do not explain satisfactorily the experimental observations, and are not appropriate for describing motion at a microscopic scale. During the first third of the 20th century, a new mechanics was developed -named “quantum mechanics” by Max Born in 1924-which governs the laws of motion for microscopic bodies but that coincides with classical mechanics when applied to macroscopic bodies.
Quantum mechanics is often presented as a brand-new development based on axioms, some of them counterintuitive, and which seem to be hardly related to classical mechanics and other precedent theories. The axiomatic approach has indeed methodological advantages, but overlooks the fact that quantum theory is intimately related with classical mechanics, specially with its Hamiltonian formulation. Bohr’s theory of electronic orbits employed Hamiltonian methods when it was realized the importance of separable systems in the formulation of the quantum conditions of Sommerfeld and Wilson in 1916; and also in the calculations of the Stark effect performed by Epstein the same year.
The reinterpretation of the quantum laws given by Schrödinger, Heisenberg and Dirac also emanated from Hamiltonian methods. The matrix nature of the canonical variables (q, p) was introduced by Heisenberg, Born and Jordan; while Dirac considered the conjugated variables as non-commutative operators. On the other hand, Schrödinger developed the operational point of view, and going farther than the analogy between optics and mechanics already established by Hamilton, turned the classical Hamilton-Jacobi equation into a wave equation. There is, therefore, a passage going from classical mechanics to quantum mechanics through optics, a path taken by Schrödinger one century after Hamilton.
In 1831 William Rowan Hamilton imagined the analogy between the trajectory of material particles moving in potential fields and the path of light rays in media with continuously variable refractive indices. Because of its great mathematical beauty, the “Hamiltonian analogy” survived in textbooks of dynamics for almost a hundred years, but did not stimulate any practical application until 1925, when H. Busch explained the focusing effect of the electromagnetic field on electron beams in optical terms, which would inspire the quick development of electronic microscopy from 1928 onwards. Almost simultaneously, in 1926, Erwin Schrödinger went one step beyond and, using the ideas presented by Louis de Broglie in 1924, moved from geometric optics to wave optics of particles . In the present work we remember and reconstruct this passage.
A typical mechanical system, to which many physical systems are reduced, consists in a collection of mass points interacting between them following well-known laws. Experience shows that the state of the system is entirely determined by the assembly of positions and velocities of all its particles. The reference system chosen to describe positions and velocities needs not to be Cartesian: the representation of the system can be achieved by means of generalized coordinates, q = (q 1 , q 2 , • • • , q n ), and generalized velocities, q = ( q1 , q2 , • • • , qn ). The minimum number, n, of generalized coordinates which are needed to describe completely the state of the system is called the number of degrees of freedom.
The laws of mechanics are those that determine the motion of the system by providing the time evolution of the positions, q(t), and the velocities , q(t), of all its components. Perhaps the most general and concise way of stating these laws is through a variational principle known as the principle of least action or Hamilton’s principle:
where the symbol δ stands for an infinitesimal variation of the trajectory. Here the Lagrangian, L = L(q, q, t), is a function of the generalized coordinates and velocities of every particle of the system. It is a general characteristic of the equations of classical physics that they can be derived from variational principles. Some examples are Fermat’s principle in optics (1657) and Maupertuis’principle in mechanics (1744). Equations from elasticity, hydrodynamics and electrodynamics can also be derived in the same way. The variational principle expressed by (1) means that when the system moves from some initial configuration at a given time t 1 to a final configuration at some other time t 2 , the real trajectory is such that the action in
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