Physlib.QuantumMechanics.HarmonicOscillator.Basic

For a quantum harmonic oscillator $Q$ with mass $m$, the mass is strictly positive, i.e., $m > 0$.

For a quantum harmonic oscillator $Q$, the mass $m$ of the particle is non-negative, satisfying $m \ge 0$.

For a quantum harmonic oscillator, the mass $m$ of the particle is non-zero ($m \neq 0$).

For a quantum harmonic oscillator $Q$, the natural frequency $\omega_i$ associated with the $i$-th dimension is strictly positive, satisfying $\omega_i > 0$.

For a quantum harmonic oscillator, the natural frequency $\omega_i$ in the $i$-th dimension is non-negative, satisfying $0 \le \omega_i$.

For a quantum harmonic oscillator $Q$, the natural frequency $\omega_i$ associated with the $i$-th dimension is non-zero.

For a quantum harmonic oscillator $Q$ with mass $m$ and natural frequency $\omega_i$ in the $i$-th dimension, the characteristic length $\xi_i$ is defined as: $$\xi_i = \frac{\sqrt{\hbar}}{\sqrt{m} \sqrt{\omega_i}}$$ where $\hbar$ is the reduced Planck's constant.

For a quantum harmonic oscillator $Q$ with mass $m$ and natural frequency $\omega_i$ in the $i$-th dimension, the characteristic length $\xi_i$ is given by the expression: $$\xi_i = \frac{\sqrt{\hbar}}{\sqrt{m} \sqrt{\omega_i}}$$ where $\hbar$ is the reduced Planck's constant.

For a quantum harmonic oscillator $Q$, the characteristic length $\xi_i$ in the $i$-th dimension is strictly positive, i.e., $$\xi_i > 0$$ where $\xi_i = \frac{\sqrt{\hbar}}{\sqrt{m} \sqrt{\omega_i}}$.

For a quantum harmonic oscillator $Q$, the characteristic length $\xi_i$ in the $i$-th dimension is non-negative, satisfying $0 \le \xi_i$.

For a quantum harmonic oscillator $Q$, the characteristic length $\xi_i$ in the $i$-th dimension is non-zero, denoted by $\xi_i \neq 0$.

For a quantum harmonic oscillator with mass $m$ and natural frequency $\omega_i$ in the $i$-th dimension, the square of the characteristic length $\xi_i$ is given by the formula: $$\xi_i^2 = \frac{\hbar}{m \omega_i}$$ where $\hbar$ is the reduced Planck's constant.

For a quantum harmonic oscillator $Q$ with mass $m$ and natural frequency $\omega_i$ in the $i$-th dimension, the inverse of the characteristic length $\xi_i$ is given by: $$\xi_i^{-1} = \frac{\sqrt{m} \sqrt{\omega_i}}{\sqrt{\hbar}}$$ where $\hbar$ is the reduced Planck's constant.

For a quantum harmonic oscillator with mass $m$ and natural frequency $\omega_i$ in the $i$-th dimension, the reciprocal of the characteristic length $\xi_i$ satisfies the relation $$\xi_i^{-1} = \frac{m \omega_i \xi_i}{\hbar}$$ where $\hbar$ is the reduced Planck's constant.

The potential matrix $M$ of a $d$-dimensional harmonic oscillator is a $d \times d$ real diagonal matrix whose $i$-th diagonal entry is $\frac{1}{2} m \omega_i^2$, where $m$ is the mass of the particle and $\omega_i$ is the natural frequency in the $i$-th dimension.

For a $d$-dimensional harmonic oscillator $Q$ with mass $m$ and natural frequencies $\omega$, the potential matrix $M$ is equal to the diagonal matrix whose $i$-th diagonal entry is $\frac{1}{2} m \omega_i^2$. That is, $$M = \operatorname{diag}\left(\frac{1}{2} m \omega_1^2, \dots, \frac{1}{2} m \omega_d^2\right).$$

The potential matrix $M$ of a $d$-dimensional harmonic oscillator is Hermitian. This matrix $M$ is a $d \times d$ real diagonal matrix where the diagonal elements are given by $M_{ii} = \frac{1}{2} m \omega_i^2$ for $i \in \{1, \dots, d\}$, where $m$ is the mass of the particle and $\omega_i$ is the natural frequency in the $i$-th dimension.

For a $d$-dimensional harmonic oscillator with mass $m$ and natural frequencies $\omega \in \mathbb{R}^d$, let $M$ be the potential matrix. For any vector $\mathbf{v} \in \mathbb{R}^d$, the matrix-vector product $M \mathbf{v}$ is given by: $$M \mathbf{v} = \frac{1}{2} m (\omega^2 \odot \mathbf{v})$$ where $\omega^2$ is the vector of squared frequencies $(\omega_1^2, \dots, \omega_d^2)$ and $\odot$ denotes the element-wise (Hadamard) product of vectors.

The potential quadratic form of a $d$-dimensional harmonic oscillator is the quadratic form $V : \mathbb{R}^d \to \mathbb{R}$ associated with the potential matrix $M$. For any vector $v \in \mathbb{R}^d$, the form is defined as $V(v) = v^\top M v$. Given the definition of the potential matrix, this evaluates to $V(v) = \sum_{i=1}^d \frac{1}{2} m \omega_i^2 v_i^2$, where $m$ is the mass of the particle and $\omega_i$ are the natural frequencies in each dimension.

The potential function $V: \text{Space } d \to \mathbb{R}$ of a $d$-dimensional harmonic oscillator is the function that maps a position $\mathbf{x}$ to its potential energy. It is defined as the composition of the potential quadratic form with the coordinate valuation map. For a particle of mass $m$ and natural frequencies $\omega_i$ for $i \in \{1, \dots, d\}$, the potential is given by: $$V(\mathbf{x}) = \frac{1}{2} m \sum_{i=1}^d \omega_i^2 x_i^2$$

For a $d$-dimensional quantum harmonic oscillator $Q$, the potential function $V: \text{Space } d \to \mathbb{R}$ is equal to the composition of the potential quadratic form $\mathcal{V}: \mathbb{R}^d \to \mathbb{R}$ and the coordinate valuation map $\text{val}: \text{Space } d \to \mathbb{R}^d$. That is, $V = \mathcal{V} \circ \text{val}$.

For a $d$-dimensional quantum harmonic oscillator, the potential function $V: \text{Space } d \to \mathbb{R}$ is almost everywhere (a.e.) strongly measurable.

A harmonic oscillator $Q$ is said to be isotropic if all its natural frequencies $\omega_i$ are equal for every dimension, such that $\omega_i = \omega_j$ for all indices $i$ and $j$.

A quantum harmonic oscillator $Q$ is isotropic if and only if all its natural frequencies are equal, such that $\omega_i = \omega_j$ for all indices $i$ and $j$.

For any quantum harmonic oscillator $Q$ in one dimension, $Q$ is isotropic. A harmonic oscillator is defined as isotropic if all its natural frequencies $\omega_i$ are equal for every dimension $i$.

The Hilbert space for a quantum harmonic oscillator in $d$ dimensions is defined as the space $L^2(\text{Space } d, \mathbb{C})$ of square-integrable complex-valued functions on the $d$-dimensional configuration space $\text{Space } d$.

For a $d$-dimensional quantum harmonic oscillator $Q$ with mass $m$ and Hilbert space $\mathcal{H} = L^2(\text{Space } d, \mathbb{C})$, the **kinetic energy operator** $\hat{T} : \mathcal{H} \to \mathcal{H}$ is a partially defined linear operator defined by scaling the momentum-squared operator $\mathbf{p}^2$ by the reciprocal of twice the mass: $$\hat{T} = \frac{1}{2m} \mathbf{p}^2$$ In terms of the Laplacian operator $\Delta$, the action of $\hat{T}$ on a wave function $\psi$ in its domain (typically the Schwartz space) is given by $\hat{T}\psi = -\frac{\hbar^2}{2m} \Delta \psi$.

For a $d$-dimensional harmonic oscillator $Q$ with Hilbert space $\mathcal{H} = L^2(\text{Space } d, \mathbb{C})$, the **potential operator** $\hat{V} : \mathcal{H} \to \mathcal{H}$ is the partially defined linear operator that acts as multiplication by the real-valued potential function $V(\mathbf{x}) = \frac{1}{2} m \sum_{i=1}^d \omega_i^2 x_i^2$. It maps a wave function $\psi(\mathbf{x})$ to the product $V(\mathbf{x})\psi(\mathbf{x})$.

For a quantum harmonic oscillator system, the potential operator $\hat{V}$ acting on the Hilbert space $\mathcal{H}$ is self-adjoint.

For a $d$-dimensional quantum harmonic oscillator $Q$ with Hilbert space $\mathcal{H} = L^2(\text{Space } d, \mathbb{C})$, the **Hamiltonian operator** $\hat{H} : \mathcal{H} \to \mathcal{H}$ is the partially defined linear operator over $\mathbb{C}$ defined as the sum of the kinetic energy operator $\hat{T}$ and the potential energy operator $\hat{V}$: $$\hat{H} = \hat{T} + \hat{V}$$

For a $d$-dimensional quantum harmonic oscillator $Q$, the Hamiltonian operator $\hat{H}$ is the sum of the kinetic energy operator $\hat{T}$ and the potential energy operator $\hat{V}$: $$\hat{H} = \hat{T} + \hat{V}$$

The Hamiltonian operator $H$ for a quantum harmonic oscillator is essentially self-adjoint on the Hilbert space $\mathcal{H}$ of the system.

The $d$-dimensional harmonic oscillator is defined as a quantum system. This system is characterized by a Hilbert space and a self-adjoint Hamiltonian operator $\hat{H}$, which is constructed as the sum of the kinetic energy operator $\hat{T}$ and the potential energy operator $\hat{V}$ (the quadratic potential).