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8: Central Potentials

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    In this chapter, we shall investigate the interaction of a non-relativistic particle of mass \(m\) and energy \(E\) with various so-called central potentials, \(V(r)\), where \(r=(x^2+y^2+z^2)^{1/2}\) is the radial distance from the origin. It is, of course, most convenient to work in spherical coordinates—\(r\), \(\theta\), \(\phi\)—during such an investigation. (See Section [s8.3].) Thus, we shall be searching for stationary wavefunctions, \(\psi(r,\theta,\phi)\), that satisfy the time-independent Schrödinger equation (see Section [sstat])

    \[ H\,\psi = E\,\psi, \label{e9.1} \]

    where the Hamiltonian takes the standard non-relativistic form

    \[ H = \frac{p^2}{2\,m} + V(r). \label{e9.2} \]

    • 8.1: Derivation of Radial Equation
      This page details the representation of momentum in Cartesian and spherical coordinates, establishing the connection between momentum and angular momentum through the anti-symmetric tensor and commutation relations. It provides an expression for angular momentum squared \(L^2\) and discusses its implications with non-commuting operators.
    • 8.2: Infinite Spherical Potential Well
      This page explores the behavior of a particle of mass \(m\) in a central potential. It describes the wavefunction, which is zero at boundaries and well-behaved at the origin, leading to a second-order differential equation with spherical Bessel functions as solutions. Energy levels are quantized based on the zeros of these functions, determined by quantum numbers \(n\), \(l\), and \(m\). The orthogonality of the resulting wavefunctions is emphasized.
    • 8.3: Hydrogen Atom
      This page covers the hydrogen atom as a two-body system of an electron and proton, which can be simplified using reduced mass. It derives the radial wavefunction from the Schrödinger equation and discusses its behavior at different distances. The wavefunction is a product of radial and angular components, with normalization conditions and expectation values based on quantum numbers.
    • 8.4: Rydberg Formula
      This page explains that electrons in hydrogen can stay in stationary states indefinitely but can transition to other states when perturbed, such as through photon interactions. Energy changes during these transitions are quantified by \(\Delta E = E_0(1/n_f^2 - 1/n_i^2)\), with signs indicating photon absorption or emission.
    • 8.E: Central Potentials (Exercises)
      This page covers quantum mechanics topics including potential wells, particularly a finite spherical well and conditions for ground states. It discusses the three-dimensional harmonic oscillator solved in spherical coordinates and explores the wavefunction for hydrogen-like atoms, deriving essential constants and energy values.

    This page titled 8: Central Potentials is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.