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arXiv:quant-ph/9904102 · Journal of Physics A Mathematical and General

Semiclassical dynamics of a spin-½ in an arbitrary magnetic field

Adrian Alscher, Hermann Grabert

Published Jan 1, 1999• 17 citations
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Abstract

The spin coherent state path integral describing the dynamics of a spin-½ system in a magnetic field of arbitrary time dependence is considered. Defining the path integral as the limit of a Wiener regularized expression, the semiclassical approximation leads to a continuous minimal action path with jumps at the endpoints. The resulting semiclassical propagator is shown to coincide with the exact quantum mechanical propagator. A nonlinear transformation of the angle variables allows for a determination of the semiclassical path and the jumps without solving a boundary-value problem. The semiclassical spin dynamics is thus readily amenable to numerical methods.

DOI
10.1088/0305-4470/32/26/309

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Comment on `Semiclassical dynamics of a spin-½ in an arbitrary magnetic field'

E. A. Kochetov

In a recent paper Alscher and Grabert claim to prove that a dominant (tree-level) stationary phase approximation to the Wiener regularized continuum su(2) coherent-state path integral for the quantum propagator : = ⟨zF | T exp (-i∫0τH ds ) | zI⟩ becomes exact, provided H is a linear combination of the su(2) generators with arbitrary time-dependent coefficients. I find the derivation of this in fact obvious result unduly complicated and somewhat obscure. The authors start from a classical spin action inconsistent with necessary boundary conditions and therefore are forced to invoke a nontrivial regularization of the action to render the latter meaningful. Alternatively, when a su(2) symplectic potential consistent with the boundary conditions is employed, no regularization is required to obtain the leading quasiclassical asymptotics.

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The semiclassical propagator for spin coherent states

Michael Stone, Kee-Su Park, Anupam Garg

We use a continuous-time path integral to obtain the semiclassical propagator for minimal-spread spin coherent states. We pay particular attention to the “extra phase” discovered by Solari and Kochetov, and show that this correction is related to an anomaly in the fluctuation determinant. We show that, once this extra factor is included, the semiclassical propagator has the correct short time behavior to O(T2), and demonstrate its consistency under dissection of the path.

arXiv:cond-mat/0004247•Journal of Mathematical Physics•Dec 1, 2000•84 citations
The spin coherent state path integral describing the dynamics of a spin-½ system in a magnetic field of arbitrary time dependence is considered. Defining the path integral as the limit of a Wiener regularized expression, the semiclassical approximation leads to a continuous minimal action path with jumps at the endpoints. The resulting semiclassical propagator is shown to coincide with the exact quantum mechanical propagator. A nonlinear transformation of the angle variables allows for a determination of the semiclassical path and the jumps without solving a boundary-value problem. The semiclassical spin dynamics is thus readily amenable to numerical methods.
In a recent paper Alscher and Grabert claim to prove that a dominant (tree-level) stationary phase approximation to the Wiener regularized continuum su(2) coherent-state path integral for the quantum propagator : = ⟨zF | T exp (-i∫0τH ds ) | zI⟩ becomes exact, provided H is a linear combination of the su(2) generators with arbitrary time-dependent coefficients. I find the derivation of this in fact obvious result unduly complicated and somewhat obscure. The authors start from a classical spin action inconsistent with necessary boundary conditions and therefore are forced to invoke a nontrivial regularization of the action to render the latter meaningful. Alternatively, when a su(2) symplectic potential consistent with the boundary conditions is employed, no regularization is required to obtain the leading quasiclassical asymptotics.
We present a semiclassical method for simulating the dynamics of nonadiabatic transitions in a discrete-state quantum system coupled to a bath of explicit continuous coordinates. This method employs a coherent-state formulation of the path integrals for the discrete system whose dynamics is described by spin operators. This spin coherent-state formulation allows the discrete system to be mapped onto a continuous coordinate. Stationary approximations of the resulting coherent-state path integrals of the system plus bath lead to quasiclassical equations of motion which can be solved numerically by direct integration. This algorithm reduces the problem to a number of simple classical trajectory calculations and does not require calculating any fluctuation determinants.
A continuous path integral is used to obtain the semiclassical expressions of quantum propagators by means of coherent states in the holomorphic representation together with the appropriate boundary conditions. It allows us to express the total actions including boundary terms, classical trajectories, and extra phases in the stationary phase approximation. In particular, in the semiclassical expression for the quantum spin propagator, we pay attention to the extra quantum phase discovered by Solari and Kochetov. It is shown that this extra phase is related to an anomaly in the fluctuation determinant. It is verified that once this extra phase is included, the semiclassical propagator leads to the correct short time behaviour to O(T²), and shows its consistency under dissection of the path. In the application to the instanton methods, the extra quantum phase yields the correct formula for the tunnel splitting including both the exponential factor and the prefactor. The quantum phase leads to the correct spectrum with a shift of the spin quantum number by 1/2. We apply the semiclassical propagators and the tunnel splitting to a few models in magnetic molecular systems. It is shown that the formula gives the correct exponential factor and prefactors for quantum tunneling of a large spin if the extra quantum phase is taken into account.