PhyslibAlpha.QuantumMechanics.HarmonicOscillator.Vacuum

For any vector $x \in \text{Space } d$ and any coordinate index $i$, the partial derivative of the real-valued Gaussian function $f(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$ with respect to the $i$-th coordinate is given by: $$\partial_i \exp\left(-\frac{1}{2} \|x\|^2\right) = -x_i \exp\left(-\frac{1}{2} \|x\|^2\right)$$ where $\|x\|$ is the Euclidean norm and $x_i$ is the $i$-th component of $x$.

For any vector $x$ in the $d$-dimensional real inner product space $\text{Space } d$, the pointwise value of the standard Gaussian Schwartz function $\mathcal{G} \in \mathcal{S}(\text{Space } d, \mathbb{C})$ is given by the real-valued Gaussian $$\mathcal{G}(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$$ where $\|x\|$ is the Euclidean norm on $\text{Space } d$.

Let $\Omega: \text{Space } d \to \mathbb{C}$ be the standard Gaussian function defined by $\Omega(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$. For any point $x \in \text{Space } d$ and any coordinate index $i \in \{0, \dots, d-1\}$, the partial derivative of $\Omega$ with respect to the $i$-th coordinate satisfies: $$\partial_i \Omega(x) = -x_i \Omega(x)$$ where $x_i$ denotes the $i$-th component of the vector $x$.

Let $\Omega \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be the standard Gaussian function defined by $\Omega(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$. For any index $j \in \{0, \dots, d-1\}$, the $j$-th component of the momentum operator $\hat{p}_j$ acting on $\Omega$ satisfies $$(\hat{p}_j \Omega)(x) = i \hbar x_j \Omega(x)$$ for all $x \in \mathbb{R}^d$, where $\hbar$ is the reduced Planck constant, $i$ is the imaginary unit, and $x_j$ is the $j$-th coordinate of the position vector $x$.

For a $d$-dimensional harmonic oscillator $Q$, `diagEquiv` is the continuous linear equivalence from $\text{Space } d$ to itself that performs a diagonal rescaling of coordinates. For any point $x \in \text{Space } d$ and coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the transformed vector is given by $(Q.\text{diagEquiv } x)_i = \xi_i x_i$, where $\xi_i$ is the characteristic length of the oscillator in the $i$-th dimension.

Let $Q$ be a $d$-dimensional harmonic oscillator. The diagonal rescaling operator $Q.\text{diagEquiv} : \text{Space } d \to \text{Space } d$ is defined such that for any vector $x \in \text{Space } d$ and any coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the transformed vector is given by $$(Q.\text{diagEquiv } x)_i = \xi_i x_i$$ where $\xi_i$ is the characteristic length of the oscillator in the $i$-th dimension and $x_i$ is the $i$-th component of the vector $x$.

Let $Q$ be a $d$-dimensional harmonic oscillator with characteristic lengths $\xi_i$ for $i \in \{0, \dots, d-1\}$. Let $Q.\text{diagEquiv}^{-1}$ be the inverse of the diagonal rescaling map. For any vector $x \in \text{Space } d$, the $i$-th component of the transformed vector is given by: $$(Q.\text{diagEquiv}^{-1} x)_i = \xi_i^{-1} x_i$$ where $x_i$ denotes the $i$-th component of $x$.

For a $d$-dimensional harmonic oscillator $Q$ with characteristic lengths $\xi_i$ for $i \in \{0, \dots, d-1\}$, the vacuum Gaussian is the complex-valued Schwartz function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$ defined by: $$\psi(x) = \exp\left( -\frac{1}{2} \sum_{i=0}^{d-1} \left( \frac{x_i}{\xi_i} \right)^2 \right)$$ for all $x \in \text{Space } d$. This function is the anisotropic Gaussian obtained by applying the diagonal continuous linear equivalence $Q.\text{diagEquiv}$ (which rescales coordinates as $x_i \mapsto \xi_i x_i$) to the standard Gaussian $x \mapsto \exp(-\frac{1}{2}\|x\|^2)$.

For a $d$-dimensional harmonic oscillator $Q$ and any vector $x \in \text{Space } d$, the value of the vacuum Gaussian function $\psi_Q(x)$ is given by $$\psi_Q(x) = \exp\left( -\frac{1}{2} \|Q.\text{diagEquiv}^{-1}(x)\|^2 \right)$$ where $Q.\text{diagEquiv}^{-1}$ is the inverse of the diagonal rescaling map $x_i \mapsto \xi_i x_i$ associated with the oscillator's characteristic lengths.

For a $d$-dimensional harmonic oscillator $Q$, let $e_i$ be the $i$-th vector of the standard orthonormal basis for $\text{Space } d$, and let $\xi_i$ be the characteristic length of the oscillator in the $i$-th dimension. The inverse of the diagonal rescaling map $\text{diagEquiv}^{-1}$ acts on the basis vector $e_i$ by scaling it by the reciprocal of $\xi_i$: $$\text{diagEquiv}^{-1}(e_i) = \xi_i^{-1} e_i$$

For a $d$-dimensional harmonic oscillator $Q$ and any point $x$ in the $d$-dimensional Euclidean space $\text{Space } d$, the Fréchet derivative of the squared norm of the rescaled coordinate vector, $f(x) = \|Q.\text{diagEquiv}^{-1}(x)\|^2$, is the continuous linear map given by: $$Df(x) = (2 \langle Q.\text{diagEquiv}^{-1}(x), \cdot \rangle) \circ Q.\text{diagEquiv}^{-1}$$ where $Q.\text{diagEquiv}^{-1}$ is the inverse of the diagonal rescaling map $x_i \mapsto \xi_i x_i$, and $\langle \cdot, \cdot \rangle$ denotes the standard real inner product on $\text{Space } d$.

For a $d$-dimensional harmonic oscillator $Q$ with characteristic lengths $\xi_j$, let $g: \text{Space } d \to \mathbb{R}$ be the real-valued anisotropic Gaussian defined by $$g(x) = \exp\left( -\frac{1}{2} \sum_{j=1}^d \frac{x_j^2}{\xi_j^2} \right).$$ Then for any point $x \in \text{Space } d$ and coordinate index $i \in \{0, \dots, d-1\}$, the partial derivative of $g$ with respect to the $i$-th coordinate is: $$\partial_i g(x) = -\frac{x_i}{\xi_i^2} \exp\left( -\frac{1}{2} \sum_{j=1}^d \frac{x_j^2}{\xi_j^2} \right).$$ Note that the expression $\left\| Q.\text{diagEquiv}^{-1}(x) \right\|^2$ in the formal statement corresponds to the sum $\sum_{j=1}^d (x_j/\xi_j)^2$.

For a $d$-dimensional harmonic oscillator $Q$ with characteristic lengths $\xi_j$ for $j \in \{0, \dots, d-1\}$, let $\Omega: \text{Space } d \to \mathbb{C}$ be the vacuum Gaussian function defined by $\Omega(x) = \exp\left( -\frac{1}{2} \sum_{j=0}^{d-1} (x_j/\xi_j)^2 \right)$. For any point $x \in \text{Space } d$ and coordinate index $i$, the partial derivative of the vacuum Gaussian with respect to the $i$-th coordinate is: $$\partial_i \Omega(x) = -\frac{x_i}{\xi_i^2} \Omega(x)$$ where $x_i$ is the $i$-th coordinate of $x$ and $\xi_i$ is the characteristic length of the oscillator in that dimension.

For a $d$-dimensional harmonic oscillator $Q$, let $\Omega$ be the vacuum Gaussian state defined by $\Omega(x) = \exp\left( -\frac{1}{2} \sum_{j=0}^{d-1} (x_j/\xi_j)^2 \right)$, where $\xi_j$ are the characteristic lengths. For any coordinate index $i \in \{0, \dots, d-1\}$ and position $x$, the action of the $i$-th component of the momentum operator $\hat{p}_i$ on the vacuum Gaussian is given by: $$(\hat{p}_i \Omega)(x) = i \hbar \frac{x_i}{\xi_i^2} \Omega(x)$$ where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.

For a $d$-dimensional quantum harmonic oscillator $Q$ with characteristic lengths $\xi_j$, let $\hat{a}_i$ be the annihilation operator for the $i$-th mode and let $\psi_Q$ be the vacuum Gaussian function defined by $$\psi_Q(x) = \exp\left( -\frac{1}{2} \sum_{j=0}^{d-1} \left( \frac{x_j}{\xi_j} \right)^2 \right)$$ for $x \in \mathbb{R}^d$. Then for any $i \in \{0, \dots, d-1\}$, applying the annihilation operator to the vacuum Gaussian yields zero: $$\hat{a}_i \psi_Q = 0.$$

For a $d$-dimensional harmonic oscillator $Q$ with characteristic lengths $\xi_i$ for $i \in \{0, \dots, d-1\}$, let $\psi_Q \in \mathcal{S}(\text{Space } d, \mathbb{C})$ be the anisotropic vacuum Gaussian defined by $$\psi_Q(x) = \exp\left( -\frac{1}{2} \sum_{i=0}^{d-1} \left( \frac{x_i}{\xi_i} \right)^2 \right).$$ Then $\psi_Q$ is a vacuum state for the ladder system of $Q$, which means that for every mode $i$, the annihilation operator $\hat{a}_i$ satisfies $\hat{a}_i \psi_Q = 0$. This result holds for any choice of characteristic lengths $\xi_i$, without requiring isotropy.

Let $Q$ be a $d$-dimensional quantum harmonic oscillator. For any mode index $i \in \{0, \dots, d-1\}$, let $\xi_i$ be the characteristic length of the $i$-th mode and $\hat{a}_i$ be its corresponding annihilation operator. If $\xi_i = 1$, then applying the annihilation operator $\hat{a}_i$ to the standard Gaussian function $g(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$ yields zero: $$\hat{a}_i \left( \exp\left(-\frac{1}{2} \|x\|^2\right) \right) = 0$$

Let $Q$ be a $d$-dimensional quantum harmonic oscillator. If for every mode $i \in \{0, \dots, d-1\}$ the characteristic length $\xi_i$ is equal to $1$, then the standard Gaussian function $g(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$ (viewed as a Schwartz map in $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$) is a vacuum state for the ladder system associated with $Q$. This implies that $g(x)$ is annihilated by the annihilation operators $\hat{a}_i$ for all $i$.