Skip to main content

Mathematics for Physics

134 Potential Wells

A potential that confines a particle to a region that is used to model bound states in one or more dimensions, where a bound state is a state with discrete energy localized in the well.

The definition of a well. A potential well is a region where \(V\) is lower than in the surroundings. For energies between the bottom and the top of the walls, bound states can form. This principle is used to model electrons in boxes and nuclei in finite wells.

The infinite-square-well spectrum. The infinite square well sets \(V=0\) inside an interval and \(V=\infty\) outside, so the wavefunction vanishes at the walls. This principle is used to obtain a complete discrete spectrum.

The infinite-well energies are

\[ E_{n} = \dfrac{n^{2}\pi^{2}\hbar^{2}}{2ma^{2}},\quad n = 1, 2, 3, \ldots \]

where

  • \(a\) is the well width.
  • \(m\) is the mass.
  • \(n\) is the quantum number.
  • \(\hbar\) is the reduced Planck constant.

The finite number of bound states in a finite well. A finite well supports only a finite number of bound states. This principle is used to contrast the infinite well, which supports infinitely many discrete energies.

Note: Also called a potential well.

134.1 References

  1. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — potential well and infinite square well.
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(E_{n}=\dfrac{n^{2}\pi^{2}\hbar^{2}}{2ma^{2}}\).
  3. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — finite versus infinite wells.