Quantum mechanics on the half-line using path integrals

Clark, T. J. H.; Menikoff, Ralph; Sharp, David H.

doi:10.1103/physrevd.22.3012

Cited by 79 publications

(97 citation statements)

References 2 publications

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“…In general, different boundary conditions require different definitions of the measure. (The measures corresponding to semi-infinite and finite line segments with boundary conditions appropriate for quantum mechanics have been discussed by Clark et al (1980); Farhi and Gutmann (1990);and Carreau et al (1990).) To apply the path integral to the problem of general dendritic trees we must consider the measure for paths on an arbitrary branched structure with the usual boundary conditions of cable theory imposed on the membrane potential at the terminals and branching nodes.…”

Section: G(x Y T) Is Expressed As An Integral Of a Certain Mea-mentioning

confidence: 99%

The path integral for dendritic trees

1991

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We construct the path integral for determining the potential on any dendritic tree described by a linear cable equation. This is done by generalizing Brownian motion from a line to a tree. We also construct the path integral for dendritic structures with spatially-varying and/or time-dependent membrane conductivities due, for example, to synaptic inputs. The path integral allows novel computational techniques to be applied to cable problems. Our analysis leads ultimately to an exact expression for the Green's function on a dendritic tree of arbitrary geometry expressed in terms of a set of simple diagrammatic rules. These rules providing a fast and efficient method for solving complex cable problems.

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Section: G(x Y T) Is Expressed As An Integral Of a Certain Mea-mentioning

confidence: 99%

The path integral for dendritic trees

1991

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“…We keep the ordering schema of the first subsection and start with the free particle case. Form the general D-dimensional free particle Green function we consider D = 2 and obtain 18) which is logarithmically divergent for |x ′′ − x ′ | → 0 due to K 0 (z) ∝ − ln(z/2) + Ψ(1) (z → 0). Here −Ψ(1) = 0.57721 56649 01532 86061 .…”

Section: Multiple δ-Function Perturbationsmentioning

confidence: 99%

“…Considering the limit γ → −∞, i.e. making the strength of the δ-function perturbation infinitely repulsive, gives a boundary value problem with Dirichlet boundary-conditions at the boundary x = a, therefore explicitly incorporating boundary problems in the path integral [15,16,18,36,37]. Note that for a symmetrical model, i.e.…”

Section: Introductionmentioning

confidence: 99%

Path integrals for two‐ and three‐dimensional δ‐function perturbations

Grosche

1994

Annalen der Physik

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Abstract. The incorporation of two-and three-dimensional δ-function perturbations into the path-integral formalism is discussed. In contrast to the onedimensional case, a regularization procedure is needed due to the divergence of the Green-function G (V ) (x, y; E), (x, y ∈ IR 2 , IR 3 ) for x = y, corresponding to a potential problem V (x). The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for δ-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism.

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“…The important point of this section is the possibility of using quantum mechanics language to describe the elastic Brownian motion. The other interesting point is the possibility of extending this equality in to the level of path integral representation [11,12,13]. In the next section we use this correspondence to calculate different area distributions of the elastic Brownian motion.…”

Section: Elastic Brownian Motion and Quantum Mechanicsmentioning

confidence: 99%

Area distribution of an elastic Brownian motion

Rajabpour¹

2009

J. Phys. A: Math. Theor.

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We calculate the excursion and meander area distributions of the elastic Brownian motion by using the self adjoint extension of the Hamiltonian of the free quantum particle on the half line. We also give some comments on the area of the Brownian motion bridge on the real line with the origin removed. We will stress on the power of self adjoint extension to investigate different possible boundary conditions for the stochastic processes. We discuss also some possible physical applications.

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Quantum mechanics on the half-line using path integrals

Cited by 79 publications

References 2 publications

The path integral for dendritic trees

The path integral for dendritic trees

Path integrals for two‐ and three‐dimensional δ‐function perturbations

Area distribution of an elastic Brownian motion

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