Physlib.QuantumMechanics.DDimensions.Hydrogen.LaplaceRungeLenzVector

For a $d$-dimensional quantum mechanical hydrogen atom with mass $m$ and coupling constant $k$, given a regularization parameter $\varepsilon \in \mathbb{R}^\times$, the $i$-th component of the regularized Laplace-Runge-Lenz (LRL) vector operator $A_i(\varepsilon)$ is a continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ defined as: $$A_i(\varepsilon) = \frac{1}{2} \sum_{j=1}^d (p_j L_{ij} + L_{ij} p_j) - mk \frac{1}{r(\varepsilon)} x_i$$ where $p_j$ is the $j$-th component of the momentum operator, $L_{ij}$ is the $ij$-component of the angular momentum operator, $x_i$ is the $i$-th component of the position operator, and $r(\varepsilon)^{-1}$ is the regularized inverse radial distance operator.

For a $d$-dimensional quantum mechanical hydrogen atom with regularization parameter $\varepsilon \in \mathbb{R}^\times$, the square of the regularized Laplace-Runge-Lenz (LRL) vector operator, denoted as $\mathbf{A}(\varepsilon)^2$, is a continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. It is defined as the sum of the squares of the individual components of the LRL operator: $$\mathbf{A}(\varepsilon)^2 = \sum_{i=1}^d A_i(\varepsilon)^2$$ where $A_i(\varepsilon)$ is the $i$-th component of the regularized LRL vector operator.

In a $d$-dimensional quantum mechanical hydrogen atom with mass $m$ and coupling constant $k$, for a given regularization parameter $\varepsilon \in \mathbb{R}^\times$, the $i$-th component of the regularized Laplace-Runge-Lenz (LRL) operator $A_i(\varepsilon)$ can be expressed in terms of the position operators $x_j$ and momentum operators $p_j$ as: $$A_i(\varepsilon) = x_i p^2 - \left(\sum_{j=1}^d x_j p_j\right) p_i + \frac{i\hbar(d-1)}{2} p_i - mk \frac{1}{r(\varepsilon)} x_i$$ where $p^2 = \sum_{j=1}^d p_j^2$ is the momentum squared operator, $\hbar$ is the reduced Planck constant, $i$ is the imaginary unit, and $r(\varepsilon)^{-1}$ is the regularized inverse radial distance operator.

For a $d$-dimensional quantum mechanical hydrogen atom with mass $m$ and coupling constant $k$, given a regularization parameter $\varepsilon \in \mathbb{R}^\times$, the $i$-th component of the regularized Laplace-Runge-Lenz (LRL) operator $A_i(\varepsilon)$ can be expressed in terms of the angular momentum operators $L_{ij}$ and momentum operators $p_j$ as: $$A_i(\varepsilon) = \sum_{j=1}^d L_{ij} p_j + \frac{i\hbar(d-1)}{2} p_i - mk \frac{1}{r(\varepsilon)} x_i$$ where $L_{ij}$ is the $ij$-component of the angular momentum operator, $p_i$ is the $i$-th component of the momentum operator, $x_i$ is the $i$-th component of the position operator, $\hbar$ is the reduced Planck constant, $i$ is the imaginary unit, and $r(\varepsilon)^{-1}$ is the regularized inverse radial distance operator.

For a $d$-dimensional quantum mechanical hydrogen atom with mass $m$ and coupling constant $k$, given a regularization parameter $\varepsilon \in \mathbb{R}^\times$, the $i$-th component of the regularized Laplace-Runge-Lenz (LRL) operator $A_i(\varepsilon)$ can be expressed in terms of the momentum operators $p_j$ and angular momentum operators $L_{ij}$ as: $$A_i(\varepsilon) = \sum_{j=1}^d p_j L_{ij} - \frac{i\hbar(d-1)}{2} p_i - mk \frac{1}{r(\varepsilon)} x_i$$ where $p_j$ is the $j$-th component of the momentum operator, $L_{ij}$ is the $ij$-component of the angular momentum operator, $x_i$ is the $i$-th component of the position operator, $\hbar$ is the reduced Planck constant, $i$ is the imaginary unit, and $r(\varepsilon)^{-1}$ is the regularized inverse radial distance operator.

For a $d$-dimensional quantum mechanical hydrogen atom, let $L_{ij}$ be the component of the angular momentum operator and $A_k(\varepsilon)$ be the $k$-th component of the regularized Laplace-Runge-Lenz operator with regularization parameter $\varepsilon \in \mathbb{R}^\times$. For any indices $i, j, k \in \{1, \dots, d\}$, the commutator between these operators satisfies: $$\left[L_{ij}, A_k(\varepsilon)\right] = i\hbar (\delta_{ik} A_j(\varepsilon) - \delta_{jk} A_i(\varepsilon))$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck's constant, and $\delta$ is the Kronecker delta.

For a $d$-dimensional quantum mechanical hydrogen atom, let $L_{ij}$ be the $ij$-component of the angular momentum operator and $\mathbf{A}(\varepsilon)^2$ be the square of the regularized Laplace-Runge-Lenz operator with regularization parameter $\varepsilon \in \mathbb{R}^\times$. For any indices $i, j \in \{1, \dots, d\}$, the commutator between these operators vanishes: $$\left[L_{ij}, \mathbf{A}(\varepsilon)^2\right] = 0$$ where $[\cdot, \cdot]$ denotes the commutator bracket.

For a $d$-dimensional quantum mechanical hydrogen atom, let $L^2$ be the square of the angular momentum operator and $\mathbf{A}(\varepsilon)^2$ be the square of the regularized Laplace-Runge-Lenz operator with regularization parameter $\varepsilon \in \mathbb{R}^\times$. The commutator of these two operators vanishes: $$\left[L^2, \mathbf{A}(\varepsilon)^2\right] = 0$$

For a $d$-dimensional quantum mechanical hydrogen atom with mass $m$ and coupling constant $k$, given a regularization parameter $\varepsilon \in \mathbb{R}^\times$, the commutator of the $i$-th and $j$-th components of the regularized Laplace-Runge-Lenz (LRL) vector operator $A_i(\varepsilon)$ and $A_j(\varepsilon)$ is given by: $$\left[A_i(\varepsilon), A_j(\varepsilon)\right] = \left(-2i\hbar m H(\varepsilon) + i\hbar mk \varepsilon^2 r(\varepsilon)^{-3}\right) L_{ij}$$ where $H(\varepsilon)$ is the regularized Hamiltonian operator, $L_{ij}$ is the $ij$-component of the angular momentum operator, $\hbar$ is the reduced Planck constant, $i$ is the imaginary unit, and $r(\varepsilon)^{-3}$ is the regularized inverse cubed radial distance operator.

For a $d$-dimensional quantum mechanical hydrogen atom with coupling constant $k$, given a regularization parameter $\varepsilon \in \mathbb{R}^\times$ and a component index $i$, the commutator of the regularized Hamiltonian $H(\varepsilon)$ and the $i$-th component of the regularized Laplace-Runge-Lenz (LRL) operator $A_i(\varepsilon)$ is given by: $$[H(\varepsilon), A_i(\varepsilon)] = i\hbar k \varepsilon^2 r(\varepsilon)^{-3} p_i - \frac{3}{2} \hbar^2 k \varepsilon^2 r(\varepsilon)^{-5} x_i$$ where $\hbar$ is the reduced Planck's constant, $p_i$ is the $i$-th component of the momentum operator, $x_i$ is the $i$-th component of the position operator, and $r(\varepsilon)^{-n}$ denotes the operator corresponding to the regularized inverse $n$-th power of the radial distance.

For a $d$-dimensional quantum mechanical hydrogen atom with mass $m$ and coupling constant $k$, given a regularization parameter $\varepsilon \in \mathbb{R}^\times$, the square of the regularized Laplace-Runge-Lenz (LRL) vector operator $\mathbf{A}(\varepsilon)^2$ is related to the regularized Hamiltonian $H(\varepsilon)$ and the square of the angular momentum operator $\mathbf{L}^2$ by the following identity: $$\mathbf{A}(\varepsilon)^2 = 2m H(\varepsilon) \left( \mathbf{L}^2 + \frac{\hbar^2 (d-1)^2}{4} \right) + m^2 k^2 (1 - \varepsilon^2 r(\varepsilon)^{-2}) - \frac{1}{2} (d-1) m k \hbar^2 \varepsilon^2 r(\varepsilon)^{-3}$$ where $\hbar$ is the reduced Planck constant and $r(\varepsilon)$ is the regularized radial distance. The operators are understood to act on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.