The study of bound states in a two-body (2B) system, consisting of two particles with reduced mass \(\mu\), begins with the Schrödinger equation. This equation, which governs the quantum behavior of the system, is expressed as:\[(H_0+V)|\Psi \rangle = E|\Psi\rangle,\]where \(H_0\) represents the free Hamiltonian,\(V\) denotes the interaction potential between the two particles, \(E\) is the binding energy of the 2B system, and \(|\Psi\rangle\) is the wavefunction describing the bound state of the system.

Rearranging the Schrödinger equation, one obtains the Lippmann-Schwinger (LS) equation, a powerful framework for solving scattering and bound state problems in quantum mechanics:\[|\Psi\rangle = \frac{1}{E - H_0} V |\Psi\rangle.\]This formulation explicitly expresses the bound-state wavefunction \(|\Psi\rangle\) as an iterative solution dependent on the interaction potential \(V\) and the free Hamiltonian \(H_0\).

In momentum space, the projection of the LS equation takes the form:\[\begin{aligned}\psi ({\bf p}) = \frac1{E - \dfrac{p ^ 2}{2 \mu}}\int_0^{\infty} dp' \, p' \int_0^{2\pi} d\phi' \, V({\bf p}, {\bf p}') \psi({\bf p}') \label{eq.LS_vector}, \end{aligned}\]where \(\psi({\bf p})\) is the momentum-space wavefunction, and \(V({\bf p}, {\bf p})\) represents the interaction potential in momentum space. Here, the completeness relation of the 2B momentum basis states is used:\[\begin{aligned}\int d^2p \, \vert {\bf p}\rangle \langle {\bf p}\vert \equiv \int_0^\infty dp \, p \int_0^{2pi} d\phi \, \vert {\bf p}\rangle \langle {\bf p}\vert = 1. \end{aligned}\]

To simplify the problem further, a Partial Wave (PW) representation is often employed, where the angular dependence of the wave function is expanded in terms of angular momentum eigenstates. This reduces the 2D integral equation into a simpler radial form:\[\begin{aligned}\psi_m (p) = \frac1{E - \frac{p ^ 2}{2\mu}} \int_0^{\infty} dp' \, p' \, V_m (p, p') \, \psi_m (p'), \label{eq.LS_pw}\end{aligned}\]where the potential in the PW representation,\(V_m(p, p')\), is obtained through the following integral: \[\begin{aligned}V_m(p, p') = \int_0^{2pi} d\phi' \, V(p, p', \phi') \cos(m \phi') \equiv \int_0^\infty dr \, J_m(pr) V(r) J_m(p'r). \end{aligned}\]Here, \(J_m(x)\)denotes the \(m\)th order Bessel function of the first kind, which arises naturally in 2D problems due to the circular symmetry of the system.

The geometry of the problem is illustrated in Fig. [Fig_2B_coordinate], where the relative momenta \({\bf p}\) and \({\bf p}'\) are described in polar coordinates. The angle\(\phi'\) corresponds to the azimuthal angle between the vectors \({\bf p}\) and \({\bf p}'\):

The PW representation significantly simplifies solving the LS equation by reducing the dimensionality of the integral and isolating contributions from each angular momentum channel \(m\). This method is particularly effective in analyzing systems with rotational symmetry, such as those encountered in 2D quantum mechanics.

In summary, the Lippmann-Schwinger equation provides a robust framework for describing 2B bound states in 2D systems. The use of the PW representation and appropriate symmetry considerations enables an efficient solution of the integral equation, shedding light on the nature of the interactions and the resulting bound states.

Numerical Implementation

To numerically solve the Lippmann-Schwinger equation [eq.LS_pw] for 2B bound states in 2D, represented in the Partial Wave (PW) framework, the integral equation should be transformed into a discretized form that can be solved iteratively or through matrix diagonalization.

To efficiently handle the integrals, variable transformations are introduced to map the infinite domains of momentum \(p\), spatial coordinate \(r\), and angle \(\phi\) onto finite intervals. The transformations are defined as: \[p = \frac{1 + x}{1 - x}, \quad r = \frac{1 + x}{1 - x}, \quad \phi = \pi (1+x), \quad x \in [-1, 1],\] where \(x\) is a variable spanning the interval\([-1, 1]\), suitable for integration using Gaussian-Legendre quadrature. The discretized form of the integral equation for the wave function \(\psi_m\) becomes an eigenvalue problem:\[\lambda \cdot \psi_m = {\cal K}(E) \, \psi_m,\] where \({\cal K}(E)\)represents the integral kernel operator dependent on the binding energy\(E\). The eigenvalue \(\lambda\) is used as a criterion to verify the solution.

Determining the Binding Energy

The binding energy \(E\) is determined by solving for the eigenvalue \(\lambda = 1\). This involves iteratively searching over initial guesses for \(E\) until convergence is achieved. A tolerance of \(10^-6\) is set for the eigenvalue condition, ensuring high accuracy in the computed energy. The iterative process can be summarized as follows:

1. Choose an initial guess for the binding energy \(E\).

2. Construct the kernel \({\cal K}(E)\) based on the chosen energy.

3. Solve the eigenvalue problem \(\lambda \cdot \psi_m = {\cal K}(E) \psi_m\) to find \(\lambda\) and the corresponding wave function \(\psi_m\).

4. Adjust \(E\) iteratively until\(\lambda\) converges to 1 within the specified tolerance.

The kernel \({cal K}(E)\) is diagonalized at each step, which allows for the extraction of both the eigenvalue \(\lambda\) and the wave function \(\psi_m\). Efficient numerical techniques such as matrix diagonalization, implemented through libraries like LAPACK, or iterative solvers like the Lanczos algorithm, can be employed for this purpose.

Numerical Results

Pedagogical and Numerical Test: Hydrogen Atom in 2D

The study of the hydrogen atom in two dimensions (2D) serves as a pedagogical example for understanding bound states and wave functions in quantum mechanics. Unlike its three-dimensional counterpart, the 2D hydrogen atom exhibits unique energy quantization and wave function characteristics due to the reduced spatial dimensions. This section explores the analytical and numerical solutions for the system.

• Particle Properties: The mass of the particle is normalized to\(m = 1\) for simplicity, allowing the focus to remain on the interaction and quantum behavior.

• Two-Body Potential: The interaction potential in real space is given by: \[V(r) = -\dfrac2{r},\]which translates to a momentum-space representation as:\[V({\bf p}, {\bf p}') = \dfrac{-1}{\pi \vert {\bf q}\vert}, \quad {\bf q}= {\bf p}- {\bf p}'.\]This form of the potential reflects the reduced dimensionality and ensures proper Coulomb-like behavior in 2D.

• Bound State Energy Levels: The energy levels for the two-body (2B) bound states in 2D are derived analytically and are given by:\[E_n = -\dfrac1{\left(n + 1/2\right)^2}, \quad n = 0, 1, 2, \ldots\] These quantized energy levels highlight the discrete nature of bound states in this system, with the energy becoming less negative as the quantum number \(n\) increases.

Pedagogical and Numerical Test: Deuteron in 2D

This section details the numerical simulation of a deuteron (the bound state of one proton and one neutron) in two dimensions using the Malfliet-Tjon (MT) potential, highlighting the adaptations made for reduced dimensions and their implications on binding energy calculations. The simulation parameters are specifically adjusted to reflect realistic physical interactions within a 2D framework.

• The mass of proton and neutron is given as \(\hbar^2/m = 41.47\) MeV fm\(^2\).

• Malfliet-Tjon potential in configuration space: The potential is expressed as a sum of attractive and repulsive Yukawa interactions (exponential terms scaled by distance \(r\)): \[V(r) = V_r \frac{e ^ {-\mu_r r}}{r} + V_a \frac{e ^ {-\mu_a r}}{r},\]where \(V_r\) and \(V_a\) are the strengths of the repulsive and attractive components, respectively, and \(\mu_r\) and \(\mu_a\) are their corresponding range parameters.

• Malfliet-Tjon potential in momentum space: In momentum space, the potential transforms to: \[V({\bf p}, {\bf p}') = \frac1{2\pi}\left(\frac{V_r}{\sqrt{q ^ 2 + \mu_r^2}} + \frac{V_a}{\sqrt{q ^ 2 + \mu_a^2}} \right), \quad q = | {\bf p}- {\bf p}' |,\]

Parameters of two models of MT potential used in our calculations. Hadi check parameters!

MT-model	\(V_a \, \text{(MeV fm)}\)	\(\mu_a \, (\text{fm}^{-1})\)	\(V_r \, \text{(MeV fm)}\)	\(\mu_r \, (\text{fm}^{-1})\)
Model-1	\(-600.00\)	\(1.550\)	\(1438.7228\)	\(3.21\)
Model-2	\(-600.00\)	\(1.550\)	\(0\)	\(0\)

Calculated deuteron binding energy in 2D for both models of the MT Potential. These calculations utilized 200 mesh points for the magnitude of two-body relative momenta and 101 mesh points for angular variables.

\(E_{d}\) (MeV)		\(E^0_{d}\) (MeV)	\(E^1_{d}\) (MeV)
Model-1		Model-2
\(-6.246\)		\(-7806\)	\(-312.4\)