Physlib.QuantumMechanics.DDimensions.Operators.AngularMomentum

For a given dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the $ij$-th component of the angular momentum operator is a continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to itself, defined by the commutator-like expression $L_{ij} = x_i p_j - x_j p_i$. Here, $x_i$ denotes the $i$-th component of the position operator and $p_j$ denotes the $j$-th component of the momentum operator.

The notation $\mathbf{L}[i, j]$ denotes the $ij$-th component of the angular momentum operator acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. For indices $i, j \in \{0, \dots, d-1\}$, it is defined as $\mathbf{L}_{ij} = \mathbf{x}_i \mathbf{p}_j - \mathbf{x}_j \mathbf{p}_i$, where $\mathbf{x}_i$ and $\mathbf{p}_j$ are the position and momentum operators, respectively.

For any dimension $d$, indices $i, j \in \{0, \dots, d-1\}$, and Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the action of the $ij$-th component of the angular momentum operator $\mathbf{L}_{ij}$ on $\psi$ is given by: $$\mathbf{L}_{ij} \psi = \mathbf{x}_i (\mathbf{p}_j \psi) - \mathbf{x}_j (\mathbf{p}_i \psi)$$ where $\mathbf{x}_i$ and $\mathbf{p}_j$ denote the $i$-th component of the position operator and the $j$-th component of the momentum operator, respectively.

For any dimension $d$, indices $i, j \in \{0, \dots, d-1\}$, Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, and point $x \in \mathbb{R}^d$, the value of the $ij$-th component of the angular momentum operator $\mathbf{L}_{ij}$ applied to $\psi$ at the point $x$ is given by: $$(\mathbf{L}_{ij} \psi)(x) = x_i (\mathbf{p}_j \psi)(x) - x_j (\mathbf{p}_i \psi)(x)$$ where $x_i$ and $x_j$ are the $i$-th and $j$-th coordinates of $x$, respectively, and $\mathbf{p}_k$ denotes the $k$-th component of the momentum operator.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the $ij$-th component of the angular momentum operator $L_{ij}$ satisfies the antisymmetry relation $L_{ij} = -L_{ji}$.

For any dimension $d$ and index $i \in \{0, \dots, d-1\}$, the component of the angular momentum operator with repeated indices, denoted as $L_{ii}$, is the zero operator acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

For a dimension $d$, the angular momentum squared operator $\mathbf{L}^2$ is a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ defined by $$\mathbf{L}^2 = \frac{1}{2} \sum_{i,j=0}^{d-1} L_{ij} \circ L_{ij},$$ where $L_{ij}$ represents the $ij$-th component of the angular momentum operator.

The symbol $\mathbf{L}^2$ denotes the total angular momentum squared operator $\text{angularMomentumOperatorSqr}$. It acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ and is defined as the sum $\mathbf{L}^2 = \frac{1}{2} \sum_{i,j=1}^d \mathbf{L}_{ij} \circ \mathbf{L}_{ij}$, where $\mathbf{L}_{ij}$ are the components of the angular momentum operator.

For any dimension $d$ and any wavefunction $\psi$ in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the action of the angular momentum squared operator $\mathbf{L}^2$ on $\psi$ is given by the sum $$\mathbf{L}^2 \psi = \frac{1}{2} \sum_{i=0}^{d-1} \sum_{j=0}^{d-1} L_{ij}(L_{ij} \psi),$$ where $L_{ij}$ denotes the $ij$-th component of the angular momentum operator.

For any dimension $d$, wavefunction $\psi$ in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, and position $x \in \mathbb{R}^d$, the value of the wavefunction $\mathbf{L}^2 \psi$ at the point $x$ is given by the double sum: $$(\mathbf{L}^2 \psi)(x) = \frac{1}{2} \sum_{i=0}^{d-1} \sum_{j=0}^{d-1} (L_{ij}(L_{ij} \psi))(x),$$ where $\mathbf{L}^2$ is the angular momentum squared operator and $L_{ij}$ is the $ij$-th component of the angular momentum operator.

In one dimension ($d=1$), for any indices $i, j \in \{0\}$, the $ij$-th component of the angular momentum operator $L_{ij}$ is the zero operator acting on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$.

The angular momentum (pseudo)scalar operator in two dimensions is the continuous linear map $L: \mathcal{S}(\mathbb{R}^2, \mathbb{C}) \to \mathcal{S}(\mathbb{R}^2, \mathbb{C})$ defined by the component $L_{01} = x_0 p_1 - x_1 p_0$, where $x_i$ and $p_j$ are the position and momentum operators respectively.

For an index $i \in \{0, 1, 2\}$, the $i$-th component of the three-dimensional angular momentum (pseudo)vector operator is a continuous linear map $L_i: \mathcal{S}(\mathbb{R}^3, \mathbb{C}) \to \mathcal{S}(\mathbb{R}^3, \mathbb{C})$ defined by the relation $L_i = \frac{1}{2} \sum_{j,k=0}^2 \epsilon_{ijk} L_{jk}$, where $\epsilon_{ijk}$ is the Levi-Civita symbol and $L_{jk} = x_j p_k - x_k p_j$ are the components of the angular momentum operator. Specifically, the components are: - $L_0 = L_{12} = x_1 p_2 - x_2 p_1$ - $L_1 = L_{20} = x_2 p_0 - x_0 p_2$ - $L_2 = L_{01} = x_0 p_1 - x_1 p_0$