The Radial Part Of The Solution of the 2-dimensional Schrödinger Equation with Circular Symmetry

Consider a two-dimensional system with a radially symmetric potential, specifically a particle of mass $m$ in a circularly symmetric potential $V(r)$. In polar coordinates $(r, \theta)$, the time-independent Schrödinger equation for energy $E$ is:

$$V(r) \psi(r,\theta) - \frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} \right) \psi(r,\theta) = E \psi(r,\theta)$$

Assuming a separable solution:

$$\psi(r,\theta) = R(r) \Theta(\theta)$$

With the angular part given by:

$$\Theta(\theta) = e^{i \theta m_l }$$

This choice for the angular solution is based on the fortunate periodic boundary conditions of the angular coordinate, ensuring solutions are periodic in $\theta$. For $m_l = 0$, this becomes $\Theta(\theta) = 1$, indicating no angular dependence.

Given this, and considering the case of a free particle with $V(r) = 0$, the radial part of the equation simplifies to:

$$- \frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r} \right) R(r) = E R(r)$$

Further simplifying with a substitution $u(r) = rR(r)$ and moving both terms to the same side:

$$E u(r) + \frac{\hbar^2}{2m} \ddot{u}(r) = 0$$

This equation has the form of Bessel's differential equation of order $0$, leading to solutions in terms of the Bessel functions of the first kind of order zero:

$$u(r) = A J_0\left( \sqrt{\frac{2mE}{\hbar^2}} r \right)$$

Here:

• $A$ is a normalization constant.
• $J_0$ represents the Bessel function of the first kind of order zero.
• $m$ is the mass of the particle.
• $E$ is the energy of the particle.
• $\hbar$ is the reduced Planck constant, which connects the energy of a wave to its frequency.
• $m_l$ is the angular quantum number, which defines the angular part of the wave function. For $m_l = 0$, the particle has zero angular momentum.

This expression gives the radial part of the wavefunction for a free particle in two dimensions, with circular symmetry and zero angular momentum. It's crucial to apply appropriate boundary conditions, especially when exploring scenarios beyond the free particle case, to determine the complete set of eigenstates.