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Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences

I.—On Hamilton's Principal Function in Quantum Mechanics

Edmund Taylor Whittaker

Published Jan 1, 1941• 1 citation
Publisher

Abstract

The solution of a problem in classical dynamics can be described, as Hamilton showed in 1834, by its Principal Function. Considering for simplicity a conservative problem with one degree of freedom, let the co-ordinate at the instant t be q, and let the Lagrangean function be L. Let Q be the value of the co-ordinate at a previous instant T. Let the quantity , after the integration has been performed, be expressed in terms of (q, Q, t — T), and let the function thus obtained, which is Hamilton's Principal Function, be denoted by W.

DOI
10.1017/s0080454100006026

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The solution of a problem in classical dynamics can be described, as Hamilton showed in 1834, by its Principal Function. Considering for simplicity a conservative problem with one degree of freedom, let the co-ordinate at the instant t be q, and let the Lagrangean function be L. Let Q be the value of the co-ordinate at a previous instant T. Let the quantity , after the integration has been performed, be expressed in terms of (q, Q, t — T), and let the function thus obtained, which is Hamilton's Principal Function, be denoted by W.
We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure. Quanta 2022; 11: 42–52.
A well-known formulation of classical mechanics of system of particles is based on solving the Hamilton-Jacobi equation ∂S ∂t + H ( qi , ∂S ∂qi , t ) = 0, (19.1) where S(qi , αi , t) is Hamilton’s principal function, H is the Hamiltonian, qi are the configuration space variables, αi are constants of integration, and t denotes time. If the Hamiltonian is independent of t and is a constant of motion equal to the total energy E, then S and Hamilton’s characteristic function W are related by S = W − Et . (19.2) Since W is independent of time, the surfaces with W = constant have fixed locations in configuration space. However, the S = constant surfaces move in time and may be considered as wavefronts propagating in this space [1]. For a single particle one can show that the wave velocity at any point is given by u = E|∇W | = E p = E mv . (19.3) T&F Cat # K11224, Chapter 19, Page 302, 15-9-2010 This shows that the velocity of a point on this surface is inversely proportional to the spatial velocity of the particle. Also one can show that the trajectories of the particle must always be normal to the surfaces of constant S. The momentum of the particle is obtained as [1] p = ∇W . (19.4) It is not clear whether this wave picture was known to classical physicists. However, in hindsight, we may now observe that there exists a correspondence between classical mechanics and geometrical optics. That is, the particle trajectories orthogonal to surfaces of constant S in classical mechanics, as we have discussed above, are similar to light rays traveling orthogonal to Huygens’ wavefronts.
The canonical functional action in the path integral in phase space is discretized by linking each pair of consecutive vertebral points - and or and - through the invariant complete solution of the Hamilton-Jacobi equation associated with the classical path defined by these extremes. When the measure is chosen to reflect the geometrical character of the propagator (it must behave as a density of weight in both of its arguments), the resulting infinitesimal propagator is cast in the form of an expansion in a basis of short-time solutions of the wave equation, associated with the eigenfunctions of the initial momenta canonically conjugated to a set of normal coordinates. The operator ordering induced by this prescription is a combination of a symmetrization rule coming from the phase, and a derivative term coming from the measure.
A dynamical treatment of a quantum in a potential is proposed by using the formulation of quantum mechanics with classical pictures and defining the quantum mechanical (abbreviated to q. m.) principal function. It is shown in the case of Coulomb potential that the q. m. momentum with some parameters can be uniquely determined and the q. m. principal function can be obtained in principle. On this occasion a new dynamical postulate is introduced. In addition to the above, it is qualitatively demonstrated that the q. m. quantities in the classical region are approximately equal to the classical ones and are larger than those in the non-classical region. Accordingly we may consider the effect of the so-called quantum potential to be small in comparison with that of the usual potential.
A dynamical treatment of a quantum in a potential is proposed by using the formulation of quantum mechanics with classical pictures and defining the quantum mechanical (abbreviated to q. m.) principal function. It is shown in the case of Coulomb potential that the q. m. momentum with some parameters can be uniquely determined and the q. m. principal function can be obtained in principle. On this occasion a new dynamical postulate is introduced. In addition to the above, it is qualitatively demonstrated that the q. m. quantities in the classical region are approximately equal to the classical ones and are larger than those in the non-classical region. Accordingly we may consider the effect of the so-called quantum potential to be small in comparison with that of the usual potential.