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.