PhyslibAlpha.QuantumMechanics.HarmonicOscillator.LadderOperators

For a $d$-dimensional quantum harmonic oscillator $Q$, the annihilation operator for the $i$-th mode, denoted as $\hat{a}_i$, is a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. It is defined by the following expression: $$\hat{a}_i = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i + \frac{i \xi_i}{\hbar} \hat{p}_i \right)$$ where $\hat{x}_i$ is the $i$-th component of the position operator, $\hat{p}_i$ is the $i$-th component of the momentum operator, $\hbar$ is the reduced Planck constant, and $\xi_i$ is the characteristic length of the $i$-th mode of the harmonic oscillator.

For a $d$-dimensional quantum harmonic oscillator $Q$ and a mode index $i \in \{0, \dots, d-1\}$, let $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be a Schwartz function. The action of the annihilation operator $\hat{a}_i$ on $\psi$ is given by the formula: $$\hat{a}_i \psi = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i \psi + \frac{i \xi_i}{\hbar} \hat{p}_i \psi \right)$$ where $\hat{x}_i$ is the $i$-th position operator, $\hat{p}_i$ is the $i$-th momentum operator, $\xi_i$ is the characteristic length of the $i$-th mode of the oscillator, and $\hbar$ is the reduced Planck constant.

For a $d$-dimensional quantum harmonic oscillator $Q$ and a mode index $i \in \{0, 1, \dots, d-1\}$, the creation operator $\hat{a}_i^\dagger$ is a continuous linear map on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ defined by: $$\hat{a}_i^\dagger = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i - \frac{i \xi_i}{\hbar} \hat{p}_i \right)$$ where $\hat{x}_i$ is the $i$-th position operator, $\hat{p}_i$ is the $i$-th momentum operator, $\xi_i$ is the characteristic length of the oscillator for the $i$-th mode, and $\hbar$ is the reduced Planck constant.

For a $d$-dimensional quantum harmonic oscillator $Q$, a mode index $i \in \{0, 1, \dots, d-1\}$, and a Schwartz function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the action of the creation operator $\hat{a}_i^\dagger$ on $\psi$ is given by: $$\hat{a}_i^\dagger \psi = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i \psi - \frac{i \xi_i}{\hbar} \hat{p}_i \psi \right)$$ where $\hat{x}_i$ is the $i$-th position operator, $\hat{p}_i$ is the $i$-th momentum operator, $\xi_i$ is the characteristic length of the oscillator for the $i$-th mode, and $\hbar$ is the reduced Planck constant.

For a $d$-dimensional quantum harmonic oscillator $Q$ and any two mode indices $i, j \in \{0, \dots, d-1\}$, the commutator of the annihilation operators $\hat{a}_i$ and $\hat{a}_j$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is zero: $$[\hat{a}_i, \hat{a}_j] = 0$$

For a $d$-dimensional quantum harmonic oscillator $Q$ and any modes $i, j \in \{0, 1, \dots, d-1\}$, the commutator of the creation operators $\hat{a}_i^\dagger$ and $\hat{a}_j^\dagger$, acting as continuous linear maps on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, is zero: $$[\hat{a}_i^\dagger, \hat{a}_j^\dagger] = 0$$ where $\hat{a}_i^\dagger$ is the creation operator for the $i$-th mode.

For a $d$-dimensional quantum harmonic oscillator $Q$ and any mode indices $i, j \in \{0, \dots, d-1\}$, the commutator of the annihilation operator $\hat{a}_i$ and the creation operator $\hat{a}_j^\dagger$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ satisfies the canonical commutation relation: $$[\hat{a}_i, \hat{a}_j^\dagger] = \delta_{ij} \hat{I}$$ where $\delta_{ij}$ is the Kronecker delta and $\hat{I}$ is the identity operator on $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

For a $d$-dimensional quantum harmonic oscillator $Q$, the annihilation operators $\hat{a}_i$ and creation operators $\hat{a}_i^\dagger$ (for $i = 0, \dots, d-1$) acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ are bundled together into a `LadderSystem` structure. This structure defines the ladder operators as continuous linear maps: $$\hat{a}_i = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i + \frac{i \xi_i}{\hbar} \hat{p}_i \right), \quad \hat{a}_i^\dagger = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i - \frac{i \xi_i}{\hbar} \hat{p}_i \right)$$ where $\hat{x}_i$ is the position operator, $\hat{p}_i$ is the momentum operator, $\hbar$ is the reduced Planck constant, and $\xi_i$ is the characteristic length of the $i$-th mode. The structure also formally establishes that these operators satisfy the canonical commutation relations: $$[\hat{a}_i, \hat{a}_j^\dagger] = \delta_{ij}, \quad [\hat{a}_i, \hat{a}_j] = 0, \quad [\hat{a}_i^\dagger, \hat{a}_j^\dagger] = 0$$

For a $d$-dimensional quantum harmonic oscillator $Q$, the number operator for the $i$-th mode, denoted as $\hat{N}_i$, is a continuous linear map on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. It is defined as the composition of the creation operator $\hat{a}_i^\dagger$ and the annihilation operator $\hat{a}_i$: $$\hat{N}_i = \hat{a}_i^\dagger \hat{a}_i$$

For a $d$-dimensional quantum harmonic oscillator $Q$ and any mode $i \in \{0, \dots, d-1\}$, the number operator $N_i$ defined via the abstract ladder system structure associated with $Q$ is equal to the linear map underlying the continuous linear number operator $\hat{N}_i$ (defined as `numberCLM i`) acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

For a $d$-dimensional quantum harmonic oscillator $Q$, let $\hat{N}_i$ be the number operator for the $i$-th mode and $\hat{a}_j$ be the annihilation operator for the $j$-th mode. These operators are continuous linear maps acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. The commutator of these operators satisfies the relation: $$[\hat{N}_i, \hat{a}_j] = -\delta_{ij} \hat{a}_i$$ where $\delta_{ij}$ is the Kronecker delta.

For a $d$-dimensional quantum harmonic oscillator $Q$, let $\hat{N}_i$ be the number operator for the $i$-th mode and $\hat{a}_j^\dagger$ be the creation operator for the $j$-th mode, both acting as continuous linear maps on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. The commutator of these operators satisfies: $$[\hat{N}_i, \hat{a}_j^\dagger] = \delta_{ij} \hat{a}_i^\dagger$$ where $\delta_{ij}$ is the Kronecker delta.

For a $d$-dimensional quantum harmonic oscillator $Q$, the number operators $\hat{N}_i$ and $\hat{N}_j$ for any two modes $i, j \in \{0, \dots, d-1\}$ commute. That is, their Lie bracket (commutator) acting on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ is zero: $$[\hat{N}_i, \hat{N}_j] = 0$$