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arXiv:1906.06640

Hamilton's Dynamics in Complex Phase Space

Muhammad Adnan Shahzad

Published Jun 16, 2019• 0 citations
arXivHTMLPDF

Abstract

We present the basic formulation of Hamilton dynamics in complex phase space. We extend the Hamilton's function by including the imaginary part and find out the corresponding Hamilton's canonical equation of motion. Example of simple harmonic motion are considered and the corresponding trajectory are plotted on real and complex phase space.

DOI
10.48550/arxiv.1906.06640

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We present the basic formulation of Hamilton dynamics in complex phase space. We extend the Hamilton's function by including the imaginary part and find out the corresponding Hamilton's canonical equation of motion. Example of simple harmonic motion are considered and the corresponding trajectory are plotted on real and complex phase space.
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.
The classical limit ℏ\hbarℏ->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schrödinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schrödinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.
Quantum trajectories in complex space in the framework of the quantum Hamilton-Jacobi formalism are investigated. For time-dependent problems, the complex quantum trajectories determined from the exact analytical wave function for the free Gaussian wave packet and the coherent state in the harmonic potential are used to demonstrate that the information transported by the particles in the complex space can be used to synthesize the time-dependent wave function on the real axis. For time-independent problems, the exact complex quantum trajectories for the Eckart potential are obtained by numerically integrating the equations of motion. The unusual structure of the total potential (the sum of the classical and the quantum potentials) for the stationary states for the Eckart potential is pointed out. The variations of the complex-valued kinetic energy, classical potential, and quantum potential along the complex quantum trajectories are analyzed. This paper not only analyzes complex quantum trajectories for time-dependent and time-independent systems but also provides a unified description for complex quantum trajectories for one-dimensional problems in the quantum Hamilton-Jacobi formalism.
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.
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.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.