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

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 is true only in one or three dimensions. In general, an extra term of the form 2(n-1)(n-3)/2m4r2 has to be added to the Hamiltonian.