Phase space path integration of integrable quantum systems
The paper under review discusses a method of exactly evaluating phase space path integrals for certain quantum mechanical problems, to be called the ``method of delta functionals''. The underlying idea is that phase space path integrals with Hamiltonians linear in the coordinates q can be immediately integrated. Consider the phase space path integral \[ \int \int [{\mathcal D}p{\mathcal D}q/2\pi]\exp (i\int^{T}_{0}[p\dot q- qF(p)]dt). \] Integrating by parts in the action to isolate q, one can perform the q functional integration to get a functional [\({\mathcal D}p]\) integral involving a momentum delta functional \[ \int ([{\mathcal D}p]/2\pi)\delta (\dot p+F(p))\exp [ip(T)q(T)-ip(0)q(0)]. \] This can be evaluated to give an ordinary integral if the classical equations of motion can be solved for p in terms of t and the initial conditions. This method is often useful because point canonical transformations \(q=f(Q)\) and other transformations can convert a Hamiltonian to a form linear in the coordinates q. However, general canonical transformations cannot be used at the present stage of the paper because of the problems related to quantization which to overcome. This limits the class of the Hamiltonians to be treated with this procedure. The authors apply this method to evaluate exactly the quantum mechanical propagator for a particle in the harmonic oscillator potential, the inverse quadratic potential \(gq^{- 2}+\omega^ 2q^ 2/2,\) the Coulomb potential, the Morse potential \(e^{-2\beta q}-2\lambda e^{-\beta q},\) the \(1/\cosh^ 2q\) potential and for a free particle on the 2-sphere.
scientific article; zbMATH DE number 4209121