On the connection between the radial momentum operator and the Hamiltonian in n dimensions

TL;DR

This paper investigates the relationship between the radial momentum operator and the Hamiltonian in n-dimensional quantum mechanics, revealing an extra term needed for dimensions other than one or three.

Contribution

It demonstrates that the standard Hamiltonian-radial momentum connection requires an additional term in dimensions other than one or three.

Findings

The known Hamiltonian-radial momentum relation holds only in 1D and 3D.

An extra term proportional to 44(n-1)(n-3)/2m4r^{2}44 is necessary in general.

The extra term vanishes in 1D and 3D, confirming the standard relation.

Abstract

The radial momentum operator in quantum mechanics is usually obtained through canonical quantization of the (symmetrical form of the) classical radial momentum. We show that the well known connection between the Hamiltonian of a free particle and the radial momentum operator $\hat{H}=\hat{P}_{r}^2/2m+ $\hat{L}^2$}/2mr^{2}$ is true only in one or three dimensions. In general, an extra term of the form $\hbar^{2}(n-1)(n-3)/ 2m \cdot 4r^{2}$ has to be added to the Hamiltonian.