Physlib.QuantumMechanics.OneDimension.HarmonicOscillator.Basic

Let $f: \mathbb{R} \to \mathbb{C}$ be the function that maps a real number to its complex representation (i.e., $f(x) = x + 0i$). At any point $x \in \mathbb{R}$, the derivative of $f$ exists and is equal to $1$.

For any real number $x$, the derivative of the natural inclusion map $f: \mathbb{R} \to \mathbb{C}$ (defined by $f(x) = x + 0i$) is equal to $1$.

The inclusion map $f: \mathbb{R} \to \mathbb{C}$, which maps a real number $x$ to its complex representation $x + 0i$, is differentiable at any point $x \in \mathbb{R}$.

For a one-dimensional quantum harmonic oscillator with mass $m$ and angular frequency $\omega$, the ratio $\frac{m \omega}{2 \hbar}$ is strictly positive, where $\hbar$ is the reduced Planck's constant.

For a one-dimensional quantum harmonic oscillator with mass $m$ and angular frequency $\omega$, the expression $\frac{m \omega}{\hbar}$ is strictly positive, where $\hbar$ is the reduced Planck's constant.

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

The characteristic length $\xi$ of a one-dimensional quantum harmonic oscillator with mass $m$ and angular frequency $\omega$ is defined as $\xi = \sqrt{\frac{\hbar}{m \omega}}$, where $\hbar$ is the reduced Planck constant.

The characteristic length $\xi$ of a one-dimensional quantum harmonic oscillator is non-negative, satisfying $0 \le \xi$.

The characteristic length $\xi$ of a one-dimensional quantum harmonic oscillator, defined as $\xi = \sqrt{\frac{\hbar}{m \omega}}$ (where $\hbar$ is the reduced Planck constant, $m$ is the mass, and $\omega$ is the angular frequency), is strictly positive, i.e., $0 < \xi$.

The characteristic length $\xi$ of a one-dimensional quantum harmonic oscillator, defined as $\xi = \sqrt{\frac{\hbar}{m \omega}}$ (where $\hbar$ is the reduced Planck constant, $m$ is the mass, and $\omega$ is the angular frequency), is non-zero, i.e., $\xi \neq 0$.

For a one-dimensional quantum harmonic oscillator with mass $m$, angular frequency $\omega$, and reduced Planck constant $\hbar$, the square of the characteristic length $\xi$ is given by $$\xi^2 = \frac{\hbar}{m \omega}.$$

For a one-dimensional quantum harmonic oscillator, the absolute value of the characteristic length $\xi$ is equal to $\xi$, i.e., $|\xi| = \xi$. Here, $\xi$ is defined as $\sqrt{\frac{\hbar}{m \omega}}$, where $m$ is the mass, $\omega$ is the angular frequency, and $\hbar$ is the reduced Planck constant.

For a one-dimensional quantum harmonic oscillator with mass $m$, angular frequency $\omega$, and reduced Planck constant $\hbar$, the reciprocal of the characteristic length $\xi$ is given by $$\frac{1}{\xi} = \sqrt{\frac{m \omega}{\hbar}}.$$

For a one-dimensional quantum harmonic oscillator with mass $m$, angular frequency $\omega$, and reduced Planck constant $\hbar$, the multiplicative inverse of the characteristic length $\xi$ is given by $$\xi^{-1} = \sqrt{\frac{m \omega}{\hbar}}.$$

For a one-dimensional quantum harmonic oscillator with mass $m$, angular frequency $\omega$, and reduced Planck constant $\hbar$, the square of the reciprocal of the characteristic length $\xi$ is given by $$\left(\frac{1}{\xi}\right)^2 = \frac{m \omega}{\hbar}.$$

The notation $P^{op}$ represents the momentum operator $\hat{p}$ in the context of the one-dimensional quantum harmonic oscillator.

The notation $X^{\text{op}}$ represents the position operator within the context of a one-dimensional quantum harmonic oscillator. It is defined as a synonym for the `positionOperator`.

The Schrödinger operator for a one-dimensional harmonic oscillator with mass $m$ and angular frequency $\omega$ is an operator that maps a wavefunction $\psi: \mathbb{R} \to \mathbb{C}$ to the function defined by: $$( \hat{H} \psi )(x) = -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2}(x) + \frac{1}{2} m \omega^2 x^2 \psi(x)$$ where $\hbar$ is the reduced Planck constant and $\frac{d^2\psi}{dx^2}$ is the second derivative of $\psi$.

For a wavefunction $\psi: \mathbb{R} \to \mathbb{C}$, the Schrödinger operator $\hat{H}$ for a one-dimensional harmonic oscillator with mass $m$ and angular frequency $\omega$ is given by the formula: $$\hat{H} \psi = -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi$$ where $\hbar$ is the reduced Planck constant and $\frac{d^2 \psi}{dx^2}$ denotes the second derivative of $\psi$ with respect to the spatial coordinate.

For a wavefunction $\psi: \mathbb{R} \to \mathbb{C}$, the Schrödinger operator $\hat{H}$ of a one-dimensional quantum harmonic oscillator with mass $m$ and characteristic length $\xi$ is given by: $$(\hat{H} \psi)(x) = \frac{\hbar^2}{2m} \left( -\frac{d^2\psi}{dx^2}(x) + \frac{x^2}{\xi^4} \psi(x) \right)$$ where $\hbar$ is the reduced Planck constant, $\frac{d^2\psi}{dx^2}$ denotes the second derivative of $\psi$, and $\xi = \sqrt{\frac{\hbar}{m \omega}}$ is the characteristic length related to the angular frequency $\omega$.

For any wavefunction $\psi: \mathbb{R} \to \mathbb{C}$, the Schrödinger operator $\hat{H}$ for a one-dimensional harmonic oscillator commutes with the parity operator $\hat{P}$. That is, $$\hat{H}(\hat{P} \psi) = \hat{P}(\hat{H} \psi)$$ where the parity operator is defined as $(\hat{P}\psi)(x) = \psi(-x)$ and the Schrödinger operator is defined as $$(\hat{H} \psi)(x) = -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2}(x) + \frac{1}{2} m \omega^2 x^2 \psi(x)$$ with $m$ being the mass, $\omega$ the angular frequency, and $\hbar$ the reduced Planck constant.

Let $\psi, \phi : \mathbb{R} \to \mathbb{C}$ be two functions representing wavefunctions. If $\psi$ and $\phi$ are twice differentiable, then the Schrödinger operator $\hat{H}$ for a one-dimensional harmonic oscillator satisfies: $$\hat{H}(\psi + \phi) = \hat{H}\psi + \hat{H}\phi$$ where the Schrödinger operator is defined as $(\hat{H} \psi)(x) = -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2}(x) + \frac{1}{2} m \omega^2 x^2 \psi(x)$.

For any complex scalar $c \in \mathbb{C}$ and any wavefunction $\psi: \mathbb{R} \to \mathbb{C}$, if $\psi$ is twice differentiable, the Schrödinger operator $\hat{H}$ for a one-dimensional harmonic oscillator satisfies the homogeneity property: $$\hat{H}(c \psi) = c (\hat{H} \psi)$$ where the Schrödinger operator is defined as $$(\hat{H} \psi)(x) = -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2}(x) + \frac{1}{2} m \omega^2 x^2 \psi(x)$$ with $m$ being the mass, $\omega$ the angular frequency, and $\hbar$ the reduced Planck constant.