Physlib.QuantumMechanics.HarmonicOscillator.LadderOperators

For a quantum harmonic oscillator in $d$ dimensions, the $i$-th lowering (or annihilation) operator $a_i$ is a continuous linear map on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. It is defined by the relation: $$a_i = \frac{1}{\sqrt{2}} \left( \frac{\hat{x}_i}{\xi_i} + i \frac{\xi_i}{\hbar} \hat{p}_i \right)$$ 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 scale of the oscillator in the $i$-th dimension, and $\hbar$ is the reduced Planck constant.

For a quantum harmonic oscillator in $d$ dimensions, the $i$-th raising (or creation) operator $\hat{a}_i^\dagger$ is defined as a continuous linear map on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. Given the characteristic length $\xi_i$, the position operator $\hat{x}_i$, and the momentum operator $\hat{p}_i$, the raising operator is given 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 $\hbar$ is the reduced Planck constant and $i$ is the imaginary unit.

For a quantum harmonic oscillator in $d$ dimensions, the $i$-th lowering operator $a_i$ is expressed in terms of the $i$-th position operator $\hat{x}_i$ and the $i$-th momentum operator $\hat{p}_i$ as: $$a_i = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i + \frac{i \xi_i}{\hbar} \hat{p}_i \right)$$ where $\xi_i$ is the characteristic length of the oscillator in the $i$-th dimension, $\hbar$ is the reduced Planck constant, and $i$ in the numerator is the imaginary unit.

For a quantum harmonic oscillator in $d$ dimensions, the $i$-th raising (or creation) operator $\hat{a}_i^\dagger$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is given by the expression: $$\hat{a}_i^\dagger = \frac{1}{\sqrt{2}} \left( \frac{1}{\xi_i} \hat{x}_i - \frac{i \xi_i}{\hbar} \hat{p}_i \right)$$ 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 in the $i$-th dimension, and $\hbar$ is the reduced Planck constant.

In the field of complex numbers $\mathbb{C}$, the square of the real square root of $2$ is equal to $2$. That is, $(\sqrt{2})^2 = 2$.

For a quantum harmonic oscillator in $d$ dimensions, the commutator of the $i$-th lowering operator $a_i$ and the $j$-th raising operator $a_j^\dagger$ acting on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ is equal to the Kronecker delta $\delta_{ij}$ times the identity operator $\mathbb{1}$. That is, $$[a_i, a_j^\dagger] = \delta_{ij} \mathbb{1}$$

For a $d$-dimensional quantum harmonic oscillator, let $a_i$ and $a_j$ be the $i$-th and $j$-th lowering (annihilation) operators acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. The commutator of these operators is zero, denoted as $[a_i, a_j] = 0$.

For a quantum harmonic oscillator in $d$ dimensions, the commutator of the $i$-th and $j$-th raising (creation) operators $\hat{a}_i^\dagger$ and $\hat{a}_j^\dagger$ acting on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ is zero: $$[\hat{a}_i^\dagger, \hat{a}_j^\dagger] = 0$$

For a quantum harmonic oscillator in $d$ dimensions, the commutator of the $i$-th raising operator $a_i^\dagger$ and the $j$-th lowering operator $a_j$, acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, is given by $$[a_i^\dagger, a_j] = -\delta_{ij} \mathbb{1},$$ where $\delta_{ij}$ is the Kronecker delta and $\mathbb{1}$ is the identity operator.

For a quantum harmonic oscillator in $d$ dimensions, the $i$-th position operator $\hat{x}_i$ acting on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ can be expressed in terms of the $i$-th lowering operator $a_i$ and the $i$-th raising operator $a_i^\dagger$ as: $$\hat{x}_i = \frac{\xi_i}{\sqrt{2}} (a_i + a_i^\dagger)$$ where $\xi_i$ is the characteristic length scale of the oscillator in the $i$-th dimension.

For a quantum harmonic oscillator in $d$ dimensions, the $i$-th momentum operator $\hat{p}_i$ is given by the expression: $$\hat{p}_i = \frac{i \hbar}{\sqrt{2} \xi_i} (a_i^\dagger - a_i)$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, $\xi_i$ is the characteristic length scale for the $i$-th dimension, $a_i^\dagger$ is the $i$-th raising (creation) operator, and $a_i$ is the $i$-th lowering (annihilation) operator.

For a $d$-dimensional quantum harmonic oscillator $Q$, the $i$-th lowering (annihilation) operator $a_i$ is an unbounded linear operator on the Hilbert space $\mathcal{H} = L^2(\text{Space } d, \mathbb{C})$. The domain of this operator is the Schwartz submodule $\mathcal{S}(\text{Space } d, \mathbb{C}) \subseteq \mathcal{H}$.

For a quantum harmonic oscillator $Q$ in $d$ dimensions, the number operator $\hat{N}_i$ for the $i$-th mode is equal to the composition of the raising operator $\hat{a}_i^\dagger$ and the lowering operator $\hat{a}_i$: $$\hat{N}_i = \hat{a}_i^\dagger \hat{a}_i$$

For a $d$-dimensional quantum harmonic oscillator $Q$, let $\hat{N}_i$ be the number operator for the $i$-th mode and $a_j$ be the $j$-th lowering operator. Their commutator satisfies the following relation: $$[\hat{N}_i, a_j] = -\delta_{ij} a_j$$ where $\delta_{ij}$ is the Kronecker delta, which is $1$ if $i = j$ and $0$ otherwise.

For a $d$-dimensional quantum harmonic oscillator, let $\hat{N}_i$ be the number operator for the $i$-th mode and $\hat{a}_j^\dagger$ be the raising (creation) operator for the $j$-th mode. The commutator of these operators satisfies the relation: $$[\hat{N}_i, \hat{a}_j^\dagger] = \delta_{ij} \hat{a}_j^\dagger$$ where $\delta_{ij}$ is the Kronecker delta, which is $1$ if $i = j$ and $0$ otherwise.

For a $d$-dimensional quantum harmonic oscillator $Q$, let $\hat{N}_i$ and $\hat{N}_j$ be the number operators for the $i$-th and $j$-th modes respectively, acting as continuous linear maps on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. Then the commutator of these two operators is zero: $$[\hat{N}_i, \hat{N}_j] = 0$$

For a $d$-dimensional quantum harmonic oscillator, let $a_i$ and $a_i^\dagger$ denote the $i$-th lowering (annihilation) and raising (creation) operators, respectively, and let $N_i$ be the $i$-th number operator. These operators act as continuous linear maps on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. The composition of the lowering operator with the raising operator for the same mode $i$ satisfies the relation: $$a_i a_i^\dagger = N_i + \mathbb{1}$$ where $\mathbb{1}$ is the identity operator on the Schwartz space.

For a $d$-dimensional quantum harmonic oscillator $Q$, let $a_i$ be the $i$-th lowering (annihilation) operator and $a_i^\dagger$ be the $i$-th raising (creation) operator, both acting on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. The lowering operator $a_i$ is the formal adjoint of the raising operator $a_i^\dagger$, meaning that for all $\psi, \phi \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the following inner product relation holds: $$\langle a_i \psi, \phi \rangle = \langle \psi, a_i^\dagger \phi \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the standard $L^2$ inner product on the space of complex-valued functions over $\mathbb{R}^d$.