Quantum effects in classical systems having complex energy

Abstract

On the basis of extensive numerical studies it is argued that there are strong analogies between the probabilistic behavior of quantum systems defined by Hermitian Hamiltonians and the deterministic behavior of classical mechanical systems extended into the complex domain. Three models are examined: the quartic double-well potential $V(x)=x^4-5x^2$, the cubic potential $V(x)=frac12x^2-g{x}^3$, and the periodic potential $V(x)=-\cos x$. For the quartic potential a wave packet that is initially localized in one side of the double-well can tunnel to the other side. Complex solutions to the classical equations of motion exhibit a remarkably analogous behavior. Furthermore, classical solutions come in two varieties, which resemble the even-parity and odd-parity quantum-mechanical bound states. For the cubic potential, a quantum wave packet that is initially in the quadratic portion of the potential near the origin will tunnel through the barrier and give rise to a probability current that flows out to infinity. The complex solutions to the corresponding classical equations of motion exhibit strongly analogous behavior. For the periodic potential a quantum particle whose energy lies between -1 and 1 can tunnel repeatedly between adjacent classically allowed regions and thus execute a localized random walk as it hops from region to region. Furthermore, if the energy of the quantum particle lies in a conduction band, then the particle delocalizes and drifts freely through the periodic potential. A classical particle having complex energy executes a qualitatively analogous local random walk, and there exists a narrow energy band for which the classical particle becomes delocalized and moves freely through the potential.

16 pages, 12 figures

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• Interference in Bohmian Mechanics with Complex Action
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• Classical Trajectories for Complex Hamiltonians
• Complexified Dynamical Systems
• Spontaneous Breaking of Classical PT Symmetry
• Complex trajectory method in time-dependent WKB

Cited by in corpus (25)

• Complex Correspondence Principle
• Coupled Oscillator Systems Having Partial PT Symmetry
• PT-symmetric deformations of integrable models
• Uses of Complex Metrics in Cosmology
• Chaotic systems in complex phase space
• PT-symmetry breaking in complex nonlinear wave equations and their deformations
• Probability Density in the Complex Plane
• Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions
• How to obtain thermostatted ring polymer molecular dynamics from exact quantum dynamics and when to use it
• Thermal quantum time-correlation functions from classical-like dynamics
• Quantum tunneling from paths in complex time
• Exceptional points in quantum and classical dynamics
• Infinitely many inequivalent field theories from one Lagrangian
• Path integral derivations of novel complex trajectory methods
• Conduction bands in classical periodic potentials
• Periodic orbits for classical particles having complex energy
• Classical Particle in a Complex Elliptic Potential
• Lorentzian path integral for quantum tunneling and WKB approximation for wave-function
• Complex Adiabatic Connection: a Hidden Non-Hermitian Path from Ground to Excited States
• To Infinity and Beyond: Some ODE and PDE Case Studies
• CPT-Frames for PT-symmetric Hamiltonians
• Complexified Synchrony
• $\mathcal{P}\mathcal{T}$-Symmetry in Hartree-Fock Theory
• Path of a tunneling particle
• Effects of complex parameters on classical trajectories of Hamiltonian systems