The operator form of the effective potential governing the time evolution in n-dimensional subspace of states

TL;DR

This paper derives the operator form of the effective potential governing quantum state evolution in n-dimensional subspaces, providing explicit formulas for 2 and 3 dimensions and relating them to physical systems.

Contribution

It introduces a general formula for the effective potential in n-dimensional subspaces and explicitly applies it to 2 and 3 dimensions, connecting theory to physical systems.

Findings

Explicit formulas for effective potential in 2 and 3 dimensions

General n-dimensional formula for effective potential

Application to physical quantum systems

Abstract

This paper presents the operator form of the effective potential V governing the time evolution in 2 and 3 and n dimensional subspace of states. The general formula for the n dimensional case is considered the starting point for the calculation of the explicit formulae for 2 and 3 dimensional degenerate and non-degenerate cases. We relate the 2 and 3 dimensional cases to some physical systems which are currently investigated.