Physlib.QuantumMechanics.DDimensions.Hydrogen.Basic

For a $d$-dimensional hydrogen atom system $H$ with mass $m$, it holds that $m \neq 0$.

For a $d$-dimensional hydrogen atom characterized by mass $m$ and a potential constant $k$, the regularized Hamiltonian $\mathbf{H}(\epsilon)$ for a non-zero parameter $\epsilon \in \mathbb{R}^\times$ is a continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ defined as: $$\mathbf{H}(\epsilon) = \frac{1}{2m} \mathbf{p}^2 - k \mathbf{r}_\epsilon^{-1}$$ where $\mathbf{p}^2$ is the momentum-squared operator and $\mathbf{r}_\epsilon^{-1}$ is the regularized radius operator that acts via pointwise multiplication by the function $x \mapsto (\|x\|^2 + \epsilon^2)^{-1/2}$.

For a $d$-dimensional hydrogen atom $H$ characterized by a constant $k$, the potential function $V(x) = -k/\|x\|$ is almost everywhere measurable.

For a $d$-dimensional hydrogen atom characterized by mass $m$ and potential constant $k$, the regularized Hamiltonian $\mathbf{H}(\epsilon)$ for a non-zero parameter $\epsilon \in \mathbb{R}^\times$ is defined as a continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ such that: $$\mathbf{H}(\epsilon) = \frac{1}{2m} \mathbf{p}^2 - k \mathbf{r}(\epsilon)^{-1}$$ where $\mathbf{p}^2$ is the momentum-squared operator and $\mathbf{r}(\epsilon)^{-1}$ is the regularized radius operator, which acts via pointwise multiplication by the function $x \mapsto (\|x\|^2 + \epsilon^2)^{-1/2}$.

For a $d$-dimensional hydrogen atom $H$ characterized by mass $m$ and potential constant $k$, the regularized Hamiltonian operator $\mathbf{H}(\epsilon)$ for any non-zero real parameter $\epsilon \in \mathbb{R}^\times$ is given by the expression: $$\mathbf{H}(\epsilon) = \frac{1}{2m} \mathbf{p}^2 - k \mathbf{r}(\epsilon)^{-1}$$ where $\mathbf{p}^2 = \mathbf{p} \cdot \mathbf{p}$ is the momentum-squared operator and $\mathbf{r}(\epsilon)^{-1}$ is the regularized inverse radius operator.