Physlib.QuantumMechanics.HarmonicOscillator.NumberOperator

For a $d$-dimensional quantum harmonic oscillator $Q$ and an index $i \in \{0, \dots, d-1\}$, let $\hat{N}_i$ be the number operator of the $i$-th mode acting on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. For any two Schwartz functions $f, g \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the $L^2$ inner product satisfies: $$\langle \hat{N}_i f, g \rangle = \langle f, \hat{N}_i g \rangle$$ where the functions are viewed as elements of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$.

For a $d$-dimensional quantum harmonic oscillator $Q$, let $\hbar$ be the reduced Planck constant and $\omega_j$ be the angular frequency of the $j$-th mode. Let $N_j$ be the $j$-th number operator and $a_i$ be the $i$-th annihilation (lowering) operator, both acting as continuous linear maps on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. The commutator of the Hamiltonian $H = \sum_j \hbar \omega_j (N_j + \frac{1}{2})$ with the $i$-th annihilation operator is given by: $$\left[ \sum_j \hbar \omega_j \left( N_j + \frac{1}{2} \right), a_i \right] = -\hbar \omega_i a_i$$

In a $d$-dimensional quantum harmonic oscillator, let $\hat{N}_j$ be the number operator for the $j$-th mode, $\hat{a}_i^\dagger$ be the $i$-th raising operator, $\hbar$ be the reduced Planck constant, and $\omega_j$ be the frequency of the $j$-th mode. On the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the Hamiltonian operator is given by $$\hat{H} = \sum_{j} \hbar \omega_j \left( \hat{N}_j + \frac{1}{2} \hat{\mathbb{1}} \right),$$ where $\hat{\mathbb{1}}$ is the identity operator. The commutator of the Hamiltonian and the $i$-th raising operator satisfies: $$\left[ \hat{H}, \hat{a}_i^\dagger \right] = \hbar \omega_i \hat{a}_i^\dagger.$$

For a $d$-dimensional quantum harmonic oscillator $Q$ with angular frequencies $\omega_j$, let $\hat{N}_j$ be the number operator for the $j$-th mode acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. The Hamiltonian operator, defined as the sum $\sum_j \hbar \omega_j (\hat{N}_j + \frac{1}{2} \hat{I})$, commutes with each individual number operator $\hat{N}_i$. That is, $$\left[ \sum_{j} \hbar \omega_j \left( \hat{N}_j + \frac{1}{2} \hat{I} \right), \hat{N}_i \right] = 0$$ where $\hbar$ is the reduced Planck constant and $\hat{I}$ is the identity operator.

For a $d$-dimensional quantum harmonic oscillator $Q$ with mass $m$ and natural frequencies $\omega_j$ ($j=1, \dots, d$), let $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be a function in the Schwartz space and $x \in \mathbb{R}^d$ be a position vector. The action of the Hamiltonian operator, expressed in terms of the number operators $\hat{N}_j$, on the function $\psi$ at point $x$ satisfies: $$\left( \sum_{j} \hbar \omega_j \left( \hat{N}_j + \frac{1}{2} \hat{\mathbb{I}} \right) \psi \right)(x) = \frac{1}{2m} \sum_{i} (\hat{p}_i (\hat{p}_i \psi))(x) + V(x) \psi(x)$$ where $\hbar$ is the reduced Planck constant, $\hat{\mathbb{I}}$ is the identity operator, $\hat{p}_i$ is the $i$-th component of the momentum operator, and $V(x) = \frac{1}{2} m \sum_i \omega_i^2 x_i^2$ is the potential function of the oscillator.

For a $d$-dimensional quantum harmonic oscillator $Q$, the Schwartz submodule $\mathcal{S}(\text{Space } d, \mathbb{C})$, which consists of the space of rapidly decreasing smooth functions, is contained within the domain of the Hamiltonian operator $\hat{H}$: $$\mathcal{S}(\text{Space } d, \mathbb{C}) \subseteq \text{dom}(\hat{H})$$ where $\text{dom}(\hat{H})$ denotes the set of states in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ upon which the Hamiltonian operator is well-defined.

Let $Q$ be a $d$-dimensional quantum harmonic oscillator with angular frequencies $\omega_j$ for $j \in \{1, \dots, d\}$. For any Schwartz function $f \in \mathcal{S}(\text{Space } d, \mathbb{C})$, let $[f]$ denote its corresponding element in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$. If $[f]$ belongs to the domain of the Hamiltonian operator $\hat{H}$, then the action of $\hat{H}$ on $[f]$ is given by $$\hat{H} [f] = \left[ \sum_{j=1}^d \hbar \omega_j \left( \hat{N}_j + \frac{1}{2} \hat{\mathbb{I}} \right) f \right]$$ where $\hbar$ is the reduced Planck constant, $\hat{N}_j$ is the $j$-th mode number operator acting on the Schwartz space, and $\hat{\mathbb{I}}$ is the identity operator.

Let $Q$ be a $d$-dimensional quantum harmonic oscillator with Hamiltonian operator $\hat{H}$ and Hilbert space $\mathcal{H} = L^2(\text{Space } d, \mathbb{C})$. For any two Schwartz functions $f, g \in \mathcal{S}(\text{Space } d, \mathbb{C})$ whose corresponding elements in $\mathcal{H}$ are in the domain of $\hat{H}$, the inner product satisfies: $$\langle \hat{H} f, g \rangle = \langle f, \hat{H} g \rangle$$ This property shows that the Hamiltonian operator is symmetric when restricted to the Schwartz submodule.

For a $d$-dimensional quantum harmonic oscillator $Q$ and an index $i \in \{0, \dots, d-1\}$, the number operator $\hat{N}_i$ is defined as an unbounded linear operator on the Hilbert space $\mathcal{H} = L^2(\text{Space } d, \mathbb{C})$. Its domain is the Schwartz submodule $\mathcal{S}(\text{Space } d, \mathbb{C}) \subset \mathcal{H}$. For any state $\psi$ in this domain, the operator acts by applying the $i$-th mode's ladder operators, $\hat{N}_i \psi = \hat{a}_i^\dagger \hat{a}_i \psi$.

For a $d$-dimensional quantum harmonic oscillator $Q$ and a mode index $i \in \{0, \dots, d-1\}$, let $\hat{N}_i$ be the number operator defined as an unbounded linear operator on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$. For any state $\psi$ in the Schwartz submodule (the subspace of the Hilbert space consisting of Schwartz functions), the action of the operator is given by: $$\hat{N}_i \psi = \iota(\hat{N}_{i, \text{CLM}} (\iota^{-1}(\psi)))$$ where $\iota: \mathcal{S}(\text{Space } d, \mathbb{C}) \to \text{SchwartzSubmodule } d$ is the linear isomorphism between the Schwartz space and the Schwartz submodule of the Hilbert space, and $\hat{N}_{i, \text{CLM}}$ is the number operator acting as a continuous linear map on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$.

For a $d$-dimensional quantum harmonic oscillator $Q$ and an index $i \in \{0, \dots, d-1\}$, the number operator $\hat{N}_i$, acting as an unbounded operator on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ with the Schwartz submodule as its domain, is symmetric. This means that for any two states $\psi$ and $\phi$ in the Schwartz submodule, the inner product satisfies: $$\langle \hat{N}_i \psi, \phi \rangle = \langle \psi, \hat{N}_i \phi \rangle$$