Exploring the infinite spherical potential well
In the realm of quantum mechanics, the problem of a particle confined within an infinite spherical potential well offers fascinating insights into the fundamental nature of wavefunctions and quantization. In this blog post, I explore the mathematical nuances of this problem using spherical coordinates, focusing on the time-independent Schrödinger equation.
The theoretical setup
Consider a particle that is free to move within a sphere of radius r_0, beyond which it encounters an infinite potential barrier. The potential V(r) for this system is defined as:
V(r) = \begin{cases} 0 & \text{if } 0 \leq r \leq r_0 \\ \infty & \text{if } r > r_0 \end{cases}
This setup confines the particle strictly within the spherical boundary, requiring that the wavefunction \Psi(r,\theta,\phi) vanish at r = r_0 due to the infinite potential outside this radius.
Solving the Schrödinger equation
The Schrödinger equation in spherical coordinates is expressed as:
\left[-\frac{\hbar^2}{2m}\left(\frac{1}{r^2}\frac{\mathrm{d}}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right) + V(r)\right]\Psi(r,\theta,\phi) = E\Psi(r,\theta,\phi)
Given that V(r) = 0 within the sphere, the equation simplifies significantly, focusing primarily on the kinetic energy term and the angular momentum contributions.
Boundary conditions and radial solutions
The boundary condition at the edge of the sphere is \Psi(r_0, \theta, \phi) = 0. To satisfy this, I analyze the radial part of the wavefunction, which is governed by:
-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2 \chi}{\mathrm{d} r^2} + \left(\frac{\hbar^2}{2m}\frac{\ell(\ell+1)}{r^2}\right)\chi = E\chi
This leads to the introduction of spherical Bessel functions j_\ell(kr) for the radial component, where k = \frac{\sqrt{2mE}}{\hbar}. The allowed energy levels are quantized based on the zeros of these Bessel functions, conforming to the boundary conditions.
Spherical Harmonics
The angular portion of the problem yields spherical harmonics, characterized by the azimuthal quantum number \ell and the magnetic quantum number m. The angular equation simplifies to:
\frac{1}{Y(\theta,\phi)}\left(\frac{1}{\sin(\theta)}\frac{\partial}{\partial \theta}\left(\sin(\theta)\frac{\partial}{\partial \theta}\right) + \frac{1}{\sin^2(\theta)} \frac{\partial^2}{\partial \phi^2} \right) =\ell(\ell+1)
The solutions Y(\theta,\phi) are the product of the associated Legendre functions and the exponential function, reflecting the quantization of angular momentum.
Concluding thoughts
This exploration of the infinite spherical potential well illustrates the intricate relationship between quantum confinement, wavefunction behavior, and quantization in spherical geometries. By rigorously solving the Schrödinger equation under these boundary conditions, I have highlighted the fundamental quantum behaviors that emerge from such a confined system.
For more insights into this topic, you can find the details here.