Neural Network Emulation of the Classical Limit in Quantum Systems via Learned Observable Mappings

TL;DR

The paper addresses the problem of understanding the quantum-classical transition via the classical limit in quantum mechanics, focusing on the regime where $\hbar$ approaches zero within the framework of strict deformation quantization. It proposes a data-driven approach that learns the mapping from initial conditions and $\hbar$ to the time evolution of the observable $\langle \hat{x}(t) \rangle$ for the 1D harmonic oscillator via the Ehrenfest equations. The authors generate training data by solving the Ehrenfest equations across multiple $\hbar$ values, train a feedforward network, and demonstrate that predictions converge toward the classical trajectory as $\hbar$ decreases. This work shows that data-driven methods can complement theoretical investigations of the quantum-classical transition and provides a foundation for extending to more complex systems, including anharmonic potentials and environmental decoherence.

Abstract

The classical limit of quantum mechanics, formally investigated through frameworks like strict deformation quantization, remains a profound area of inquiry in the philosophy of physics. This paper explores a computational approach employing a neural network to emulate the emergence of classical behavior from the quantum harmonic oscillator as Planck's constant $\hbar$ approaches zero. We develop and train a neural network architecture to learn the mapping from initial expectation values and $\hbar$ to the time evolution of the expectation value of position. By analyzing the network's predictions across different regimes of hbar, we aim to provide computational insights into the nature of the quantum-classical transition. This work demonstrates the potential of machine learning as a complementary tool for exploring foundational questions in quantum mechanics and its classical limit.

Figures (2)

• Figure 1: Convergence of neural network predictions to the classical trajectory for the harmonic oscillator with varying $\hbar$. The plot shows the expectation value of position $\langle \hat{x}(t) \rangle$ vs. time for different values of $\hbar$, as predicted by the trained neural network, along with the classical trajectory ($\hbar = 0$).
• Figure 2: Zoomed-in view of the convergence plot between $t=2.0$ and $t=4.0$, illustrating the differences in predicted trajectories for various $\hbar$ values. Larger deviations from the classical trajectory are observed for higher values of $\hbar$.