Derivation of semiclassical asymptotic initial value representations of the quantum propagator

Abstract

We present a formalism based on the Bargmann (coherent state) representation of states and operators to derive asymptotic semiclassical initial value representations of the quantum propagator for general multidimensional systems. We first derive a semiclassical WKB-like approximation to the general solution of the multidimensional time dependent Schrödinger equation in the Bargmann representation. From here, we readily obtain the semiclassical asymptotic form of the coherent-state matrix elements of the propagator. This form includes terms depending on the quantization scheme chosen to quantize a classical Hamilton function or the classical symbol chosen for a given quantum Hamiltonian. From this expression and its analytic properties we derive through asymptotic saddle point approximations a whole family of semiclassical initial value representations of the quantum propagator, all of which belong to the Herman–Kluk (HK) class. A parameter in this family determines either the quantization scheme or the Hamiltonian classical symbol. The Wigner–Weyl choice for it leads to the HK propagator. Potential applications for other choices are discussed.