Physlib.QuantumMechanics.OneDimension.ReflectionlessPotential.Basic

For a quantum mechanical system in one dimension with mass $m$, reduced Planck constant $\hbar$, and parameters $\kappa$ and $N$, the reflectionless potential is the function $V: \mathbb{R} \to \mathbb{R}$ defined by $$V(x) = -\frac{\hbar^2 \kappa^2 N(N + 1)}{2m \cosh^2(\kappa x)}$$ where $x$ represents the position and $\cosh$ is the hyperbolic cosine function.

Given a real parameter $\kappa$, the `tanhOperator` is the multiplication operator that maps a function $\psi: \mathbb{R} \to \mathbb{C}$ to the function $x \mapsto \tanh(\kappa x) \psi(x)$. In the context of quantum mechanics, this represents the action of the operator $\tanh(\kappa \hat{X})$ on a wave function in the position representation.

Given a function $g: \mathbb{R} \to \mathbb{C}$ that satisfies the condition of temperate growth, this definition represents the continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ defined by pointwise multiplication by $g$. For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, the operator maps $\psi$ to the product $g \psi$.

For any real number $\kappa$, the function $x \mapsto \tanh(\kappa x)$ has temperate growth.

For any real number $\kappa$, the complex-valued function $x \mapsto \tanh(\kappa x)$ has temperate growth.

Given a reflectionless potential $Q$ with a characteristic parameter $\kappa \in \mathbb{R}$, this definition represents the continuous linear operator on the space of complex-valued Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ that acts via pointwise multiplication by the function $x \mapsto \tanh(\kappa x)$. Specifically, for any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, the operator maps $\psi$ to the product function $x \mapsto \tanh(\kappa x) \cdot \psi(x)$.

Given a reflectionless potential $Q$ characterized by mass $m$, a parameter $\kappa \in \mathbb{R}$, and the reduced Planck constant $\hbar$, the creation operator $a^\dagger$ is an operator that maps a complex-valued function $\psi: \mathbb{R} \to \mathbb{C}$ to another function defined by: \[ (a^\dagger \psi)(x) = \frac{1}{\sqrt{2m}} \left( \hat{P}\psi(x) + i \hbar \kappa \tanh(\kappa x) \psi(x) \right) \] where $\hat{P}$ is the one-dimensional momentum operator defined as $(\hat{P}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x)$, and $i$ is the imaginary unit. In operator notation, this is expressed as: \[ a^\dagger = \frac{1}{\sqrt{2m}} (\hat{P} + i \hbar \kappa \tanh(\kappa \hat{X})) \] where $\tanh(\kappa \hat{X})$ denotes the multiplication operator by the function $x \mapsto \tanh(\kappa x)$.

Given a reflectionless potential $Q$ characterized by mass $m$, a parameter $\kappa \in \mathbb{R}$, and the reduced Planck constant $\hbar$, the annihilation operator $a$ is an operator that maps a complex-valued function $\psi: \mathbb{R} \to \mathbb{C}$ to another function defined by: \[ (a\psi)(x) = \frac{1}{\sqrt{2m}} \left( (\hat{P}\psi)(x) - i \hbar \kappa \tanh(\kappa x) \psi(x) \right) \] where $\hat{P}$ is the one-dimensional momentum operator defined as $(\hat{P}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x)$, and $i$ is the imaginary unit. In operator notation, this is expressed as: \[ a = \frac{1}{\sqrt{2m}} (\hat{P} - i \hbar \kappa \tanh(\kappa \hat{X})) \] where $\tanh(\kappa \hat{X})$ denotes the multiplication operator by the function $x \mapsto \tanh(\kappa x)$.

For a reflectionless potential $Q$ characterized by mass $m$, reduced Planck constant $\hbar$, and parameter $\kappa \in \mathbb{R}$, the creation operator $a^\dagger$ is defined as the continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ given by: \[ a^\dagger = \frac{1}{\sqrt{2m}} \hat{p} + \frac{i \hbar \kappa}{\sqrt{2m}} \tanh(\kappa \hat{x}) \] where $\hat{p}$ is the momentum operator on the Schwartz space, $\tanh(\kappa \hat{x})$ is the operator of pointwise multiplication by the function $x \mapsto \tanh(\kappa x)$, and $i$ is the imaginary unit.

For a reflectionless potential $Q$ characterized by mass $m$, the reduced Planck constant $\hbar$, and a parameter $\kappa \in \mathbb{R}$, the annihilation operator $a$ is defined as the continuous linear operator on the space of complex-valued Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ given by: \[ a = \frac{1}{\sqrt{2m}} \left( \hat{p} - i \hbar \kappa \tanh(\kappa \hat{x}) \right) \] where $\hat{p}$ is the momentum operator on the Schwartz space and $\tanh(\kappa \hat{x})$ is the operator representing pointwise multiplication by the function $x \mapsto \tanh(\kappa x)$.