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Journal of Mathematical Physics

Remarks on the Hamiltonian path integral in polar coordinates

Wolfgang Langguth, Akira Inomata

Published Mar 1, 1979• 50 citations
Publisher

Abstract

Problems associated with the derivation of the Hamiltonian path integral in polar coordinates are examined. First the use of the ill-defined asymptotic formula of the modified Bessel function is pointed out. A procedure is proposed to justify its practical use, in which the mass m is complexified and the limit Imm→0 is taken after path integrations. Hereby a restriction is imposed on the class of allowed potentials. The difference between the Hamiltonian path integral so obtained and the phase space path integral formally defined is also discussed.

DOI
10.1063/1.524101

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Problems associated with the derivation of the Hamiltonian path integral in polar coordinates are examined. First the use of the ill-defined asymptotic formula of the modified Bessel function is pointed out. A procedure is proposed to justify its practical use, in which the mass m is complexified and the limit Imm→0 is taken after path integrations. Hereby a restriction is imposed on the class of allowed potentials. The difference between the Hamiltonian path integral so obtained and the phase space path integral formally defined is also discussed.
Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.
This paper defines path integrals in phase space without using a time-division approach followed by a limiting process, thereby generalizing a similar procedure used in configuration space. This is useful since the path integral approach cannot always be formulated in configuration space (e.g., when the Hamiltonian is arbitrary) but can always be formulated in phase space. The most general Gaussian measure, absorbing the quadratic portion of the functional to be integrated, is constructed, and large classes of path integrals are evaluated with respect to it. Applications are given to the perturbation expansion and the semiclassical (WKB) expansion for arbitrary Hamiltonians.
The path integral for the potential is evaluated in three different coordinate systems, i. e. in cartesian coordinates, in polar coordinates and in parabolic coordinates. The equivalence between the different approaches is shown and in each case the energyspectrum and the wave functions are explicitly calculated. Furthermore we discuss special cases, including the ring-potential, the Coulomb potential with an Aharonov-Bohm solenoid, and the genuine Coulomb problem. We also point out the separability of the path integral formulation of the ring-potential in spheroidal coordinates, and of the Coulomb potential in spheroidal and spheroconical coordinates, respectively.
Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach. The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space , defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian. To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.
The integrals in the path integral are strongly oscillating and can only be defined by analytic continuation. It is important to note that the continuous expression is just a notation for the discrete version of the path integral, but formal manipulations will be much easier to perform in this continuous formulation. Furthermore, the integral is only defined through the analytic continuation in time. As long as the Hamiltonian is quadratic in the momenta, the stationary phase approximation for the momentum integral is exact. In practical situations one splits from the action the quadratic part in the coordinates and velocities and considers the rest as a perturbation.