The paper obtains an explicit analytic expression for the radial position-momentum uncertainty product of the d-dimensional hydrogen atom in terms of the quantum numbers n and ℓ and the dimension d.
On the connection between the radial momentum operator and the Hamiltonian in n dimensions
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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.
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Generalized Radial Uncertainty Product for d-Dimensional Hydrogen Atom
The paper obtains an explicit analytic expression for the radial position-momentum uncertainty product of the d-dimensional hydrogen atom in terms of the quantum numbers n and ℓ and the dimension d.