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Physical Review A

Deriving propagators with stochastic quantum evolution

Yun‐An Yan, Jiushu Shao

Published Mar 25, 2025• 0 citations
Publisher

Abstract

Quantum evolution was recently shown to be equivalent to the mathematical expectation of the stochastic motion described by the Schr\"odinger equation with momentum-coupled white noise [J. Shao, J. Chem. Phys. 159, 024128 (2023)]. The stochastic formulation provides a convenient means of evaluating quantum propagators because the involved kinetic and potential contributions are disentangled. With the help of the auxiliary functional averaging technique, we apply this scheme to the derivation of the quantum propagators for systems of exponential, linear, harmonic, and $\ensuremath{\delta}$-function potentials. Notably, our work presents a truly workable expression of the quantum propagator for the exponential potential problem. To demonstrate the features and advantages of this propagator, we use it to calculate the wave-packet dynamics and compare it to the split-operator method.

DOI
10.1103/physreva.111.032219

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Quantum evolution was recently shown to be equivalent to the mathematical expectation of the stochastic motion described by the Schr"odinger equation with momentum-coupled white noise [J. Shao, J. Chem. Phys. 159, 024128 (2023)]. The stochastic formulation provides a convenient means of evaluating quantum propagators because the involved kinetic and potential contributions are disentangled. With the help of the auxiliary functional averaging technique, we apply this scheme to the derivation of the quantum propagators for systems of exponential, linear, harmonic, and \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potentials. Notably, our work presents a truly workable expression of the quantum propagator for the exponential potential problem. To demonstrate the features and advantages of this propagator, we use it to calculate the wave-packet dynamics and compare it to the split-operator method.
We propose a stochastic Schrödinger equation in which the momentum is coupled to a white Gaussian noise. In the stochastic representation, the kinetic energy representing the self-interaction of momentum is reduced to a linear term of momentum. As such, the quantum evolution operator factorizes into two contributions due to the momentum and the potential, respectively. The exact quantum propagator thereby becomes an expectation of the stochastic one in which the amplitude results from the potential with a fluctuating position: The particle moves with a constant velocity, subjected to a complex Brownian motion. We demonstrate that the stochastic Schrödinger equation can be feasibly used to derive the quantum propagators for the linear potential and the harmonic oscillator system. Novel semiclassical and other approximations may be developed from the new representation of quantum mechanics.
By virtue of stochastic decoupling, the quantum evolution of a harmonic oscillator is converted into two virtual contributions dictated by either a stochastic creation operator or a stochastic annihilation operator. Two approaches are provided to resolve these virtual two evolution operators. One is based on the coherent-state representation and the other deals with the momentum representation after separating out the position and momentum factors. The quantum propagator of the harmonic oscillator is obtained by taking the random average of the combined virtual propagators. The main mathematical exercise of the whole procedure is the calculation of the Gaussian integral.
The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.
The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.
We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler-Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.