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IMPERIAL COLLEGE PRESS eBooks

Hamilton-Jacobi Formulation

Published Jan 1, 2006• 0 citations
Publisher

Abstract

In the one-dimensional periodic potential case, we formulate the condition of Bloch periodicity for the reduced action by using the relation between the wave function and the reduced action established in the context of the equivalence postulate of quantum mechanics.Then, without appealing to the wave function properties, we reproduce the well-known dispersion relations which predict the band structure for the energy spectrum in the Krönig-Penney model.

DOI
10.1142/9781860949180_0005

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In the one-dimensional periodic potential case, we formulate the condition of Bloch periodicity for the reduced action by using the relation between the wave function and the reduced action established in the context of the equivalence postulate of quantum mechanics.Then, without appealing to the wave function properties, we reproduce the well-known dispersion relations which predict the band structure for the energy spectrum in the Krönig-Penney model.
Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.
It is well known in classical mechanics that the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. Analogously, the equivalent quantum action variable appearing in the quantum Hamilton–Jacobi formalism can provide the energy eigenvalues of a bound state problem, without the necessity of solving the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well-known solvable potentials. It is also shown how this method provides an understanding as to why approximate quantization schemes such as ordinary and supersymmetric WKB can give exact answers for certain potentials.
Implementation of the equivalence principle, stating that all physical systems can be connected by a coordinate transformation (trivializing map) to the free system with vanishing energy, uniquely leads to the quantum analogue of the Hamilton–Jacobi (HJ) equation. A property of the formalism is manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of the recently observed relation with second–order differential equations. This makes clear the intrinsic ψD –ψ duality between linearly independent solutions of the Schrödinger equation. Being a third–order non–linear differential equation, the Quantum Stationary HJ Equation (QSHJE) gives rise to initial conditions which cannot be seen in the Schrödinger equation. Solutions of the QSHJE provide a trajectory representation of quantum mechanics, depending on the Planck length. Unlike Bohm’s theory, there is a non– trivial action even for bound states and no wave guide is present. A fundamental aspect of the formulation is that no use of any axiomatic interpretation of the wave function is made. The QSHJE is defined only if the ratio ψD /ψ is a local homeomorphism of the extended real line into itself. In this respect the quantization of the energy is strictly related to the S duality of the Schwarzian derivative. As a result, the equivalence principle, according to which all the potentials are equivalent under coordinate transformations, by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization. Finally, we derive the quantum HJ equation also in the higher dimensional case with time dependent potentials and in the relativistic case. Contents
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.