In this work we will examine an on-shell action, its basic properties and its relation to transition amplitude. Derivatives of on-shell action with respect to position and time are equal to momentum and energy. On-shell action of a system is sufficient for determining the trajectory describing time evo- lution of the system. On-shell action can be computed as (off-shell) action of specific physical trajectory connecting initial position in initial time with final position in final time but it can also be found from solutions to two Hamilton-Jacobi equations, one in initial time and position variables and the other in final time and position variables. In quantum mechanics a transition amplitude is directly proportional to the complex exponential of the on-shell action. Work with on-shell action is demonstrated on examples such as free particle, harmonic oscillator and partly also on central force problem.
In this work we will examine an on-shell action, its basic properties and its relation to transition amplitude. Derivatives of on-shell action with respect to position and time are equal to momentum and energy. On-shell action of a system is sufficient for determining the trajectory describing time evo- lution of the system. On-shell action can be computed as (off-shell) action of specific physical trajectory connecting initial position in initial time with final position in final time but it can also be found from solutions to two Hamilton-Jacobi equations, one in initial time and position variables and the other in final time and position variables. In quantum mechanics a transition amplitude is directly proportional to the complex exponential of the on-shell action. Work with on-shell action is demonstrated on examples such as free particle, harmonic oscillator and partly also on central force problem.
A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalism a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.
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.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
First, we show that there exists in classical mechanics three actions corresponding to different boundary conditions: two well-known actions, the Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which links a family of particles of initial action S_0(x) to their various positions x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which links a particle in initial position x_0 and initial velocity v_0 to its position x at time t. We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density. In the first case, where the initial quantum density is a classical density, the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the initial density. In the second case where initial density converges to a Dirac density, the density converges to the Dirac function and the quantum action converges to a deterministic action. Therefore we introduce into classical mechanics non-discerned particles, which satisfy the statistical Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally, we propose an interpretation of the Schrodinger wave function that depends on the initial conditions (preparation). This double interpretation seems to be the interpretation of Louis de Broglie's "double solution" idea.
A quantum particle with potential energy V ( , t ) is considered in the frame of a phase-space picture of the quantum theory, and the interconnection between quantum mechanics and a -dependent extended classical dynamics is analysed. The initial position-space wavefunction determines the initial conditions for a set of Hamilton-like equations that leads up to an ensemble of complex-valued phase-space trajectories. The one-dimensional driven harmonic oscillator is used for illustrating the method, and for generating a complete set of phase-space functions.
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.