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 $V(x)=frac{1}{2}x^2-gx^3$ , 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.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 $V(x)=frac{1}{2}x^2-gx^3$, 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.

Sep 5, 200846 citations

Complex Classical Mechanics of a QES Potential*

Bhabani Prasad Mandal, Sushant S. Mahajan

Abstract We study a combined parity (P) and time reversal (T) invariant non-Hermitian quasi-exactly solvable (QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.Abstract We study a combined parity (P) and time reversal (T) invariant non-Hermitian quasi-exactly solvable (QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.

arXiv:1312.0757Communications in Theoretical PhysicsOct 1, 20158 citations

Quantum tunneling as a classical anomaly

Carl M. Bender, Daniel W. Hook

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.

arXiv:1011.0121Journal of Physics A Mathematical and TheoreticalAug 23, 201126 citations

Quantum Bound States and Constrained Random Walk?

Francesco R. Ruggeri

In quantum mechanics, one often speaks of wave-like behaviour. In a previous note, however, we argued that wavelike behaviour is based on “averaging” of complex, periodic probabilities and so represents an ensemble average taken at different times. If one does not want an average over different times, but rather follows a quantum particle in time, it seems it undergoes a constrained random walk. This random walk differs from usual statistical ones in that the stochastic hits from a potential must on average yield V(x) at each x. We argue this is the source of the complex, periodic terms exp(ipx) which are part of a Fourier series of V(x). The potential influences the momentum distribution of a particle causing it to also exhibit complex, periodic probabilities of the form exp(ipx). Thus, if one considers “or” probabilistic situations for p and -p, one may obtain real periodic functions such as sin(px) and cos(px), but we argue these are time averages. A particle is either in a state p or -p (or some other momentum state) at x at a particular time. Thus, the wave-like behaviour follows from averaging over time exp(ipx) terms, but at each instant it seems a particle follows a constrained random walk type motion possibly changing its direction of motion with each stochastic hit from a potential. Overall, however, there must be a periodicity in the bound system, but we have argued this (based on exp(iEt)) is related to changes of a time averaged ensemble. We argue that this periodicity is also found in the constrained random walk picture if one considers a series of xi points (xi) with associated momenta (pi) and uses P(xi)P(pi/xi) as the probability at each point. Then the average kinetic energy for a constrained random walk added to V(x)W(x)W(x) yields EW(x)W(x). We also argue that W(x)W(x) is related to the idea of constrained random walk. The idea of a constrained random walk in quantum mechanics has no doubt been suggested in the literature, but the main focus of quantum mechanics does not seem to be with such a random walk, but rather with averages (taken at different times) which related to a wave picture that exists only as an ensemble average.In quantum mechanics, one often speaks of wave-like behaviour. In a previous note, however, we argued that wavelike behaviour is based on “averaging” of complex, periodic probabilities and so represents an ensemble average taken at different times. If one does not want an average over different times, but rather follows a quantum particle in time, it seems it undergoes a constrained random walk. This random walk differs from usual statistical ones in that the stochastic hits from a potential must on average yield V(x) at each x. We argue this is the source of the complex, periodic terms exp(ipx) which are part of a Fourier series of V(x). The potential influences the momentum distribution of a particle causing it to also exhibit complex, periodic probabilities of the form exp(ipx). Thus, if one considers “or” probabilistic situations for p and -p, one may obtain real periodic functions such as sin(px) and cos(px), but we argue these are time averages. A particle is either in a state p or -p (or some other momentum state) at x at a particular time. Thus, the wave-like behaviour follows from averaging over time exp(ipx) terms, but at each instant it seems a particle follows a constrained random walk type motion possibly changing its direction of motion with each stochastic hit from a potential. Overall, however, there must be a periodicity in the bound system, but we have argued this (based on exp(iEt)) is related to changes of a time averaged ensemble. We argue that this periodicity is also found in the constrained random walk picture if one considers a series of xi points (xi) with associated momenta (pi) and uses P(xi)P(pi/xi) as the probability at each point. Then the average kinetic energy for a constrained random walk added to V(x)W(x)W(x) yields EW(x)W(x). We also argue that W(x)W(x) is related to the idea of constrained random walk. The idea of a constrained random walk in quantum mechanics has no doubt been suggested in the literature, but the main focus of quantum mechanics does not seem to be with such a random walk, but rather with averages (taken at different times) which related to a wave picture that exists only as an ensemble average.

Aug 3, 20200 citations

Investigation of Classical Systems with Complex Energy in the Field of Quantum-Classical Correspondence

Shahadat Hossain, M. S. Hossain

Classical mechanics and quantum mechanics contradict each other, and both are essential to explain the phenomena of the exclusively different realm of nature. On the other hand, Bohr's correspondence principle shows classical mechanics is the somewhat approximate version of quantum mechanics. Classically a particle with negative energy i. e. V E < is not allowed go through a forbidden region or disappearing from one well to another well. This paper gives the numerical studies for the trajectory of the particle in a double-well potential and presents quantum mechanical behavior such as tunneling in the complex plane for different energies. Our findings provide a route to solve the classical system with complex energy.Classical mechanics and quantum mechanics contradict each other, and both are essential to explain the phenomena of the exclusively different realm of nature. On the other hand, Bohr's correspondence principle shows classical mechanics is the somewhat approximate version of quantum mechanics. Classically a particle with negative energy i. e. V E < is not allowed go through a forbidden region or disappearing from one well to another well. This paper gives the numerical studies for the trajectory of the particle in a double-well potential and presents quantum mechanical behavior such as tunneling in the complex plane for different energies. Our findings provide a route to solve the classical system with complex energy.

Global Journal of Human Social ScienceApr 23, 20180 citations

Bloch oscillations in non-Hermitian lattices with trajectories in the complex plane

Stefano Longhi

Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called acceleration theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e., not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only on average, and that the periodic wave-packet motion of the particle in real space is described by a trajectory in the complex plane, i.e., it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called acceleration theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e., not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only on average, and that the periodic wave-packet motion of the particle in real space is described by a trajectory in the complex plane, i.e., it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.

arXiv:1510.01480Physical Review AOct 21, 201531 citations

V(x)=x^4-5x^2

, the cubic potential

V(x)=frac{1}{2}x^2-gx^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.