Measuring the Accuracy of Variational Trial Functions in Quantum Mechanics: The Road to an Accurate Description of Nodal Surfaces for Few-Electron Systems*

Since the establishment of time-independent quantum mechanics as a spectral problem [1], the variational method has played an essential role in our understanding of the quantum world. This method provides an upper bound for the lowest eigenvalue of a given Hamiltonian operator that supports bound states via a trial function. Two natural questions emerge: (i) How close is the upper bound to the exact energy? (ii) How close are the exact and trial functions? The answer to these questions can be formulated in terms of perturbation theory and its connection to the equations describing a dielectric medium [2]. These equations are solved numerically for the very first time [3]. We then discuss extensions to tackle excited states. Information about their nodal surfaces is crucial in fixed-node Monte-Carlo quantum calculations applied to few-electron systems. To exemplify the methodology, we consider a hydrogen atom in its ground state subjected to a constant magnetic field.  

[1] E. Schrödinger, Ann. Phys 386, 109 (1926).

[2] A. V. Turbiner, Sov. Phys. Usp. 27, 668 (1984). 

[3] J.C. del Valle, work in progress, (2024).