Carl M Bender et al 2008 J. Phys. A: Math. Theor. 41 352003 doi:10.1088/1751-8113/41/35/352003
Quantum effects in classical systems having complex energy
FREE ARTICLECarl M Bender1, Dorje C Brody2 and Daniel W Hook3
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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) = x4 − 5x2, the cubic potential 
, 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. Moreover, 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.
- PACS
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03.65.Ge Solutions of wave equations: bound states
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Xp Tunneling, traversal time, quantum Zeno dynamics
- MSC
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81Sxx General quantum mechanics and problems of quantization
- Subjects
- Dates
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Issue 35 (5 September 2008)
Received 25 April 2008, in final form 2 July 2008
Published 29 July 2008
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Quantum effects in classical systems having complex energy
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