Physlib.QuantumMechanics.Operators.AngularMomentum

For a given dimension $d$ and indices $i, j \in \{0, 1, \dots, d-1\}$, the $ij$-th component of the angular momentum operator $L_{ij}$ is the continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to itself defined by the relation: $$L_{ij} = \hat{x}_i \hat{p}_j - \hat{x}_j \hat{p}_i$$ where $\hat{x}_i$ is the $i$-th position operator (multiplication by the $i$-th coordinate) and $\hat{p}_j$ is the $j$-th momentum operator (acting as $-i\hbar \partial_j$ on the Schwartz space).

The definition `QuantumMechanics.term𝐋` provides the notation for the angular momentum operator components $\mathbf{L}_{ij}$. In quantum mechanics, these components are defined as $\mathbf{L}_{ij} = \mathbf{x}_i \mathbf{p}_j - \mathbf{x}_j \mathbf{p}_i$, where $\mathbf{x}_i$ is the $i$-th position operator and $\mathbf{p}_j$ is the $j$-th momentum operator acting on the space of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

This definition introduces the notation $\mathbf{L}[i, j]$ for the $(i, j)$-th component of the angular momentum operator. It corresponds to the operator defined by $\hat{L}_{ij} = x_i \hat{p}_j - x_j \hat{p}_i$ acting on the space of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, where $x_i$ is the position operator and $\hat{p}_j$ is the momentum operator.

For a dimension $d \in \mathbb{N}$ and indices $i, j \in \{0, 1, \dots, d-1\}$, let $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be a Schwartz function. The $(i, j)$-th component of the angular momentum operator $\mathbf{L}_{ij}$ applied to $\psi$ is given by: $$\mathbf{L}_{ij} \psi = \hat{x}_i (\hat{p}_j \psi) - \hat{x}_j (\hat{p}_i \psi)$$ where $\hat{x}_k$ and $\hat{p}_k$ denote the $k$-th component of the position and momentum operators, respectively.

For any dimension $d \in \mathbb{N}$ and indices $i, j \in \{0, 1, \dots, d-1\}$, let $L_{ij}$ be the component of the angular momentum operator, $\hat{x}_k$ be the $k$-th position operator, and $\hat{p}_k$ be the $k$-th momentum operator acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. For any Schwartz function $\psi$ and any point $x \in \mathbb{R}^d$, the value of the function $L_{ij} \psi$ at $x$ is given by: $$(L_{ij} \psi)(x) = (\hat{x}_i (\hat{p}_j \psi))(x) - (\hat{x}_j (\hat{p}_i \psi))(x)$$ where $(\hat{x}_k \phi)(x) = x_k \phi(x)$ is the pointwise multiplication by the $k$-th coordinate.

For any dimension $d$ and indices $i, j \in \{0, 1, \dots, d-1\}$, the components of the angular momentum operator $L_{ij}$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ satisfy the antisymmetry relation: $$L_{ij} = -L_{ji}$$ where the operator is defined by $L_{ij} = \hat{x}_i \hat{p}_j - \hat{x}_j \hat{p}_i$.

For any dimension $d$ and any index $i \in \{0, 1, \dots, d-1\}$, the $ii$-th component of the angular momentum operator, defined as $L_{ij} = \hat{x}_i \hat{p}_j - \hat{x}_j \hat{p}_i$, vanishes. That is, $L_{ii} = 0$.

The notation $\mathbf{L}^2$ represents the squared angular momentum operator. As defined in the context of quantum mechanics on a $d$-dimensional space, it acts on the space of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ according to the formula $\mathbf{L}^2 = \frac{1}{2} \sum_{i,j} \mathbf{L}_{ij} \circ \mathbf{L}_{ij}$, where $\mathbf{L}_{ij}$ are the components of the angular momentum operator.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the domain of the $ij$-th component of the angular momentum operator $L_{ij}$ is the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$, regarded as a submodule of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, let $\psi$ be a wave function belonging to the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. The $ij$-th component of the angular momentum operator $L_{ij} = x_i p_j - x_j p_i$ applied to $\psi$ is equal almost everywhere (with respect to the volume measure) to the result of applying the operator $L_{ij}$ (defined as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$) to the corresponding function in the Schwartz space.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, if a wave function $\psi$ belongs to the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, then the result of applying the $ij$-th component of the angular momentum operator $L_{ij} = x_i p_j - x_j p_i$ to $\psi$ is also an element of the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the $ij$-th component of the angular momentum operator $L_{ij} = x_i p_j - x_j p_i$, acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, has a dense domain in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the $ij$-th component of the angular momentum operator $L_{ij}$ on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ can be expressed by reversing the order of the position and momentum operators: $$L_{ij} = \hat{p}_j \hat{x}_i - \hat{p}_i \hat{x}_j$$ where $\hat{x}_i$ is the $i$-th position operator (multiplication by the $i$-th coordinate) and $\hat{p}_j$ is the $j$-th momentum operator (acting as $-i\hbar \partial_j$). This identity confirms that the definition $L_{ij} = \hat{x}_i \hat{p}_j - \hat{x}_j \hat{p}_i$ is invariant under the reversal of the operator ordering.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the $ij$-th component of the angular momentum operator $L_{ij} = x_i p_j - x_j p_i$ is a symmetric operator on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. Specifically, for any two wavefunctions $\psi, \phi$ in the Schwartz domain, the $L^2$ inner product satisfies: $$\langle L_{ij} \psi, \phi \rangle = \langle \psi, L_{ij} \phi \rangle$$ where $x_i$ and $p_j$ are the position and momentum operators, respectively.

For any dimension $d$ and indices $i, j \in \{0, \dots, d-1\}$, the $ij$-th component of the angular momentum operator $L_{ij} = x_i p_j - x_j p_i$, acting on the domain of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ within the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$, is densely defined and closable.

For a given dimension $d$ and indices $i, j \in \{0, 1, \dots, d-1\}$, let $L_{ij}$ be the $ij$-th component of the angular momentum operator acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. For any two functions $f, g \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the $L^2$ inner product satisfies the symmetry identity: $$\langle L_{ij} f, g \rangle_{L^2} = \langle f, L_{ij} g \rangle_{L^2}$$ where the Schwartz functions are treated as elements of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ via the standard inclusion.

The angular momentum squared operator $\mathbf{L}^2$ is a linear operator on the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ with the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ as its domain. It is defined by transporting the operator $\mathbf{L}^2 = \frac{1}{2} \sum_{i,j=0}^{d-1} L_{ij} \circ L_{ij}$ (where $L_{ij}$ are the components of the angular momentum operator) from the Schwartz space to the Hilbert space.

This definition introduces the notation $\mathbf{L}^2$ for the angular momentum squared operator. This operator acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ and is defined by the sum $\mathbf{L}^2 = \frac{1}{2} \sum_{i,j} \mathbf{L}_{ij} \circ \mathbf{L}_{ij}$, where $\mathbf{L}_{ij}$ are the components of the angular momentum operator.

The domain of the $d$-dimensional angular momentum squared operator $\mathbf{L}^2$ is the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

Let $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be the Schwartz space and $\mathcal{S}_{L^2} \subset L^2(\mathbb{R}^d, \mathbb{C})$ be the corresponding submodule of the Hilbert space of square-integrable functions. For any element $\psi \in \mathcal{S}_{L^2}$, the action of the angular momentum squared operator $\mathbf{L}^2$ on the Hilbert space is given by: $$\mathbf{L}^2 \psi = \Phi(\mathbf{L}^2_{\mathcal{S}}(\Phi^{-1}(\psi)))$$ where $\Phi: \mathcal{S}(\mathbb{R}^d, \mathbb{C}) \to \mathcal{S}_{L^2}$ is the canonical linear isomorphism between the Schwartz space and its image in the Hilbert space, and $\mathbf{L}^2_{\mathcal{S}}$ is the angular momentum squared operator defined on the Schwartz space.

For any dimension $d$, let $\psi$ be an element of the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. The application of the angular momentum squared operator $\mathbf{L}^2$ (as defined on the Hilbert space) to $\psi$ is equal almost everywhere (with respect to the Lebesgue volume measure) to the result of applying the Schwartz-space angular momentum squared operator to the Schwartz function corresponding to $\psi$.

For any function $\psi$ in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ (viewed as a submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$), applying the angular momentum squared operator $\mathbf{L}^2$ results in a function that is also in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

For any wave function $\psi$ in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the angular momentum squared operator $\mathbf{L}^2$ acts as the sum of the squares of the components $L_{ij}$: $$\mathbf{L}^2 \psi = \frac{1}{2} \sum_{i=0}^{d-1} \sum_{j=0}^{d-1} L_{ij}(L_{ij} \psi),$$ where $L_{ij} = x_i p_j - x_j p_i$ is the $ij$-th component of the angular momentum operator. The composition $L_{ij}(L_{ij} \psi)$ is well-defined because each $L_{ij}$ maps the Schwartz space to itself.

The domain of the angular momentum squared operator $\mathbf{L}^2$ is dense in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. Here, $\mathbf{L}^2$ is defined as a linear operator with the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ as its domain, and the dense property indicates that $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is a dense subspace of the space of square-integrable functions $L^2(\mathbb{R}^d, \mathbb{C})$.

For any two functions $f, g$ in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the angular momentum squared operator $\mathbf{L}^2$ is symmetric with respect to the $L^2$ inner product. That is, $$\langle \mathbf{L}^2 f, g \rangle = \langle f, \mathbf{L}^2 g \rangle,$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. This symmetry is obtained by shifting each component of the operator across the inner product.

The angular momentum squared operator $\mathbf{L}^2$ on the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$, with the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ as its domain, is a symmetric operator. This means that for any two functions $f, g$ in the Schwartz space, the $L^2$ inner product satisfies $\langle \mathbf{L}^2 f, g \rangle = \langle f, \mathbf{L}^2 g \rangle$.

The angular momentum squared operator $\mathbf{L}^2$ on the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$, with the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ as its domain, is densely defined and closable.