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Nuclear Physics B

Effects of barrier penetration on energy spectrum and wave functions in the path integral formalism

Iring Bender, Dieter Gromes, Heinz J. Rothe, K. D. Rothe

Published Apr 1, 1978• 14 citations
Publisher
DOI
10.1016/0550-3213(78)90281-x

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1. Gaussian integrals 2. Path integral in quantum mechanics 3. Partition function and spectrum 4. Classical and quantum statistical physics 5. Path integrals and quantization 6. Path integral and holomorphic formalism 7. Path integrals: fermions 8. Barrier penetration: semi-classical approximation 9. Quantum evolution and scattering matrix 10. Path integrals in phase space QUANTUM MECHANICS: MINIMAL BACKGROUND A1 Hilbert space and operators A2 Quantum evolution, symmetries and density matrix A3 Position and momentum. Scrodinger equation
By developing an analogy between the Feynman path integral and contour integral representations of the special functions, we obtain WKB formulas for barrier penetration from a path integral. We first show that there exists for the path integral a notion of contour independence in the time parameter. We then select an appropriate contour to describe the physical situation of barrier penetration and obtain asymptotic formulas from the function space integral. The method is interpreted as a path integral derivation of the complex ray description of barrier penetration. In the last three sections we investigate several canonical problems of the theory of complex rays with these path integral techniques.
The Aharonov–Bohm (AB) effect demonstrates that a charged particle can acquire an observable phase shift in a region where the local magnetic field identically vanishes provided the configuration space is multiply connected and the electromagnetic connection exhibits non-trivial holonomy. We present a gauge-covariant physico-mathematical framework that systematically disentangles three regimes: (i) the standard ideal AB effect governed by the holonomy of a flat but globally inexact connection on ΩR=R2∖DR with DR={r≤R}; (ii) the path-integral decomposition into homotopy sectors indexed by the winding number, which is constructed on the universal cover; and (iii) non-topological corrections arising from evanescent wave-function penetration when the solenoid is modeled as a finite radial barrier. The ideal AB phase is taken as the reference topological holonomy, while the finite-barrier extension developed here separates the measured phase, under explicit weak-penetration assumptions, as Δφtotal=ΔφABhol+δφfinite+O(Teff2). Here, ΔφABhol=qΦ/ħ is the standard topological holonomy, whereas δφfinite=O(e−2κℓb) is a small dynamical, microphysics-dependent correction controlled by the barrier height, particle energy, and effective forbidden length ℓb. Within Laskin’s fractional quantum mechanics, we use the fractional extension as a diagnostic test: the topological AB phase is independent of the Lévy index α, whereas the finite-barrier correction acquires an anomalous, α-dependent penetration length. The numerical material provides gauge-covariant diagnostics: a Peierls-substituted split-operator scheme, a sector-resolved Monte Carlo path integral benchmarked against the exact Aharonov–Bohm propagator of Gerry and Singh, a WKB/transfer-matrix finite-barrier estimate tracking δφfinite→0, and Berry-phase computations used as consistency checks.