Physlib.QuantumMechanics.PoschlTeller.Basic

The Pöschl-Teller potential function maps a position $x$ in one-dimensional space to the real value: $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$$ where $m$ is the mass of the particle, $\hbar$ is the reduced Planck's constant, $\kappa$ is a parameter controlling the width of the potential, and $N$ is a parameter determining its depth.

For a given Pöschl-Teller system, the Hilbert space is defined as $L^2(\text{Space } 1, \mathbb{C})$, the space of square-integrable complex-valued functions on one-dimensional space.

The continuous linear operator on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ defined by pointwise multiplication by the function $x \mapsto \tanh(\kappa x)$, where $\kappa$ is the characteristic parameter of the Pöschl-Teller potential.

The creation operator for a one-dimensional Pöschl-Teller system is a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ defined by the expression: $$\frac{1}{\sqrt{2m}} \left( \hat{p} + i\hbar\kappa \tanh(\kappa x) \right)$$ where $m$ is the mass of the particle, $\kappa$ is the characteristic parameter of the potential, $\hat{p}$ is the momentum operator $-i\hbar \frac{d}{dx}$, $\hbar$ is the reduced Planck constant, and $i$ is the imaginary unit. The operator involves the momentum operator and the pointwise multiplication by the hyperbolic tangent function $\tanh(\kappa x)$.

The annihilation operator $\hat{a}$ for the one-dimensional Pöschl-Teller potential is defined as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$. Given the system parameters for mass $m$ and the characteristic potential scale $\kappa$, the operator is expressed as: $$\hat{a} = \frac{1}{\sqrt{2m}} \left( \hat{p} - i\hbar\kappa \tanh(\kappa x) \right)$$ where $\hat{p}$ is the momentum operator, $\hbar$ is the reduced Planck constant, and $\tanh(\kappa x)$ denotes the operator for pointwise multiplication by the hyperbolic tangent function.

The `tanhOperator` is the unbounded linear operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ defined by the pointwise multiplication by the function $f(x) = \tanh(\kappa x)$, where $\kappa$ is the characteristic parameter of the Pöschl-Teller potential. The operator maps a wave function $\psi \in L^2(\mathbb{R}, \mathbb{C})$ to the product $(\tanh(\kappa x)) \psi(x)$ on the domain where this product remains square-integrable.

For a one-dimensional Pöschl-Teller system with mass $m$ and characteristic parameter $\kappa$, the creation operator $\hat{a}^\dagger$ is an unbounded linear operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$. It is defined as: $$\hat{a}^\dagger = \frac{1}{\sqrt{2m}} \left( \hat{p} + i\hbar\kappa \tanh(\kappa x) \right)$$ where $\hat{p}$ is the momentum operator, $\hbar$ is the reduced Planck constant, and $\tanh(\kappa x)$ denotes the operator for pointwise multiplication by the hyperbolic tangent function.

The annihilation operator for the Pöschl-Teller system is an unbounded linear operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ defined by the expression $$\frac{1}{\sqrt{2m}} (P - i \hbar \kappa \tanh(\kappa x))$$ where $P$ is the momentum operator $-i\hbar \frac{d}{dx}$, $m$ is the mass of the particle, $\hbar$ is the reduced Planck constant, and $\kappa$ is the characteristic parameter of the Pöschl-Teller potential.

For the one-dimensional Pöschl-Teller potential $V(x)$, defined as $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$$ the potential function is even, meaning that for any position $x$ in one-dimensional space, $V(-x) = V(x)$.

For the one-dimensional Pöschl-Teller system with characteristic parameter $\kappa$, the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \tanh(\kappa x)$ has temperate growth. (A function has temperate growth if it and its derivatives are bounded by some polynomial at infinity).

For the parameter $\kappa$ of the one-dimensional Pöschl-Teller potential, the complex-valued function defined by $x \mapsto \tanh(\kappa x)$ has temperate growth.

The Pöschl-Teller potential function $V: \text{Space } 1 \to \mathbb{R}$, defined as $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)},$$ has temperate growth. A function is said to have temperate growth (or be slowly increasing) if it is smooth and all its derivatives are bounded by some polynomial. Here, $m$ is the particle mass, $\hbar$ is the reduced Planck constant, and $\kappa$ and $N$ are parameters characterizing the potential's shape.

Let $V: \text{Space } 1 \to \mathbb{R}$ be the Pöschl-Teller potential defined by $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)},$$ where $m$ is the mass of the particle, $\hbar$ is the reduced Planck constant, and $\kappa$ and $N$ are parameters characterizing the potential. Then the complex-valued function $f: \text{Space } 1 \to \mathbb{C}$ defined by the composition of the real-to-complex inclusion $\text{ofReal}$ and $V$ (i.e., $f(x) = V(x) + 0i$) has temperate growth.

The Pöschl-Teller potential function $V: \text{Space } 1 \to \mathbb{R}$, defined as $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)},$$ is continuous, where $m$ is the mass of the particle, $\hbar$ is the reduced Planck constant, and $\kappa$ and $N$ are parameters characterizing the potential.

The Pöschl-Teller system is defined as a one-dimensional quantum mechanical system (a `SpaceDQuantumSystem` with dimension $d=1$). It describes a particle of mass $m$ subject to the Pöschl-Teller potential: $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$$ where $\hbar$ is the reduced Planck's constant, and $\kappa$ and $N$ are parameters determining the width and depth of the potential respectively. The structure includes the definition of the Hilbert space, the kinetic energy operator, the potential energy operator, and the total Hamiltonian $\hat{H} = \hat{T} + \hat{V}$.

For a Pöschl-Teller quantum system $Q$, the spatial dimension $d$ of its underlying $d$-dimensional quantum system structure is equal to $1$.

For a 1-dimensional Pöschl-Teller quantum system $Q$, the mass $m$ of the associated quantum mechanical system is equal to the mass $m$ defined within the Pöschl-Teller configuration.

For a one-dimensional Pöschl-Teller quantum system $Q$, the potential energy function is the Pöschl-Teller potential $V(x)$, which is defined as: $$V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$$ where $m$ is the mass of the particle, $\hbar$ is the reduced Planck's constant, $\kappa$ is the width parameter, and $N$ is the depth parameter.

For a one-dimensional Pöschl-Teller quantum system $Q$ with particle mass $m$, the kinetic energy operator $\hat{T}$ (viewed as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$) acts on a Schwartz function $\psi$ according to: $$\hat{T} \psi = \frac{1}{2m} \hat{p}(\hat{p} \psi)$$ where $\hat{p}$ is the momentum operator (the $0$-th component in one dimension).

For a one-dimensional Pöschl-Teller quantum system $Q$, the potential operator $\hat{V}$ (represented as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$) acts by pointwise multiplication. Specifically, for any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ and any position $x \in \mathbb{R}$, the value of the resulting function at $x$ is: $$(\hat{V}\psi)(x) = V(x) \psi(x)$$ where $V(x)$ is the Pöschl-Teller potential defined by $V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$.

In the one-dimensional Pöschl-Teller quantum system $Q$, the potential operator $\hat{V}$ (the multiplication operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ defined by the potential function $V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$) is self-adjoint.

For a one-dimensional Pöschl-Teller quantum system $Q$, let $\text{dom}(\hat{V})$ denote the domain of the potential operator $\hat{V}$ acting on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$. The space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ is contained within the domain of the potential operator: $$\mathcal{S}(\mathbb{R}, \mathbb{C}) \subseteq \text{dom}(\hat{V}).$$ This implies that for any Schwartz function $\psi$, the product $V \psi$ is square-integrable, where $V(x) = -\frac{\hbar^2 \kappa^2 N(N+1)}{2m \cosh^2(\kappa x)}$.

For a one-dimensional Pöschl-Teller quantum system with mass $m$, the action of the Hamiltonian operator $\hat{H}$ on a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ at a point $x \in \mathbb{R}$ is defined by the sum of the kinetic and potential energy components: $$(\hat{H}\psi)(x) = \frac{1}{2m} (\hat{p}(\hat{p}\psi))(x) + V(x)\psi(x)$$ where $\hat{p}$ is the momentum operator and $V(x)$ is the Pöschl-Teller potential function evaluated at $x$.

Let $\mathcal{S}(\mathbb{R}, \mathbb{C})$ denote the Schwartz space of complex-valued functions on the real line. For any function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ and any point $x \in \mathbb{R}$, the continuous linear operator $Q.\text{tanhCLM}$ acts on $\psi$ via pointwise multiplication by the hyperbolic tangent function: $$(Q.\text{tanhCLM} \, \psi)(x) = \tanh(\kappa x) \psi(x)$$ where $\kappa$ is the characteristic parameter associated with the Pöschl-Teller potential $Q$.

For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ and any point $x \in \mathbb{R}$, the pointwise action of the Pöschl-Teller creation operator $\hat{a}^\dagger$ is given by: $$(\hat{a}^\dagger \psi)(x) = \frac{1}{\sqrt{2m}} (\hat{p} \psi)(x) + \frac{i\hbar\kappa}{\sqrt{2m}} \tanh(\kappa x) \psi(x)$$ where $m$ is the mass of the particle, $\kappa$ is the characteristic parameter of the Pöschl-Teller potential, $\hat{p}$ is the momentum operator, $\hbar$ is the reduced Planck constant, and $i$ is the imaginary unit.

For a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ and a point $x \in \mathbb{R}$, the pointwise action of the annihilation operator $\hat{a}$ for the one-dimensional Pöschl-Teller potential with mass $m$ and characteristic scale $\kappa$ is given by: $$(\hat{a}\psi)(x) = \frac{1}{\sqrt{2m}} (\hat{p}\psi)(x) - \frac{i\hbar\kappa}{\sqrt{2m}} \tanh(\kappa x) \psi(x)$$ where $\hat{p} = -i\hbar \frac{d}{dx}$ is the momentum operator, $\hbar$ is the reduced Planck constant, and $i$ is the imaginary unit.

For a one-dimensional Pöschl-Teller system with mass $m$, the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ satisfy the following relation involving the momentum operator $\hat{p}$: $$\hat{a}^\dagger + \hat{a} = \frac{2}{\sqrt{2m}} \hat{p}$$ This means that the sum of the ladder operators is equal to the momentum operator scaled by the factor $2 / \sqrt{2m}$.

For a one-dimensional Pöschl-Teller system with mass $m$ and potential scale parameter $\kappa$, the difference between the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ is given by: $$\hat{a}^\dagger - \hat{a} = \frac{2 i \hbar \kappa}{\sqrt{2m}} \tanh(\kappa x)$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\tanh(\kappa x)$ denotes the operator for pointwise multiplication by the hyperbolic tangent function.