Summation over Feynman Histories in Polar Coordinates

Peak, David; Inomata, Akira

doi:10.1063/1.1664984

Cited by 223 publications

(106 citation statements)

References 6 publications

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“…the radial Green's function (31) can be seen to take the form of the radial Green's function for the spherical harmonic oscillator supplemented by a centrifugal barrier which has been calculated [31] with the result Hence, we have near the poles,…”

Section: The Harmonic Oscillator Plus the V (θ)−Potentialmentioning

confidence: 80%

Path integral treatment of a noncentral electric potential

et al. 2013

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Abstract:We present a rigorous path integral treatment of a dynamical system in the axially symmetric potentialIt is shown that the Green's function can be calculated in spherical coordinate system for V (θ) = . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number ≥ 0, are recovered.

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Section: The Harmonic Oscillator Plus the V (θ)−Potentialmentioning

confidence: 80%

Path integral treatment of a noncentral electric potential

et al. 2013

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“…In connection with Eq. (38) it is useful to remind that in d = 1 the discrete bound states can all be chosen to be real [32], so that the Bloch density matrix is real and symmetric and can be written in the form…”

Section: Asymptotic Behavior Of the Bloch Density Matrixmentioning

confidence: 99%

Local-time representation of path integrals

Jizba¹,

Zatloukal²

2015

Phys. Rev. E

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We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high and low temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle are also discussed.

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“…In many papers (see [2], [29] and references therein) solutions to the quantum (singular) oscillator with varying mass and/or frequency are expressed in terms of other than ǫ(t) parameter functions. Regarding the analytic solutions to the classical equation (14) for time-dependent "frequency" Ω(t), see [29,30]. In [7] the Heisenberg operators U † (t)L j U(t) =λ jk (t)L k were constructed for the initial conditions b(0) = 0 = m(0) =ω(0) =ġ(0), the coefficientsλ jk (t) being expressed in terms of a parameter functionǫ(t) which obey a slightly different second order equation.…”

Section: Symmetry and Invariants For The General Somentioning

confidence: 99%

“…As in the particular cases of b = 0 [2]- [14] we consider the collapse free case 1 + 8m(t)g(t)/h 2 = 1 + 4c ≥ 0 (c = const) and look for wave functions Ψ(x, t) which are vanishing at x = 0 (since at x → 0 the potential may tend to ∞).…”

Section: Wave Functions and Algebra Related Coherent Statesmentioning

confidence: 99%

“…Previously this singular oscillator (SO) has been treated in a number of paper [6,7,8,9,10,11,12,13,14], exact invariants and wave functions being obtained for the case of stationary SO (constant m, ω and g) in [12,14] and of SO with varying frequency ω(t) (but constant m, g) in [11,13]. The more general cases treated in [7,9] corresponds (as in [2]) to m(t)g(t) = const.…”

Section: Introductionmentioning

confidence: 99%

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Exact solutions for the general nonstationary oscillator with a singular perturbation

Trifonov¹

1999

J. Phys. A: Math. Gen.

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Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrödinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU (1, 1) group-related coherent states for the SO minimize the Robertson and Schrödinger uncertainty relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.

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Summation over Feynman Histories in Polar Coordinates

Cited by 223 publications

References 6 publications

Path integral treatment of a noncentral electric potential

Path integral treatment of a noncentral electric potential

Local-time representation of path integrals

Exact solutions for the general nonstationary oscillator with a singular perturbation

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