Particle in A Spherically Symmetric Potential - Derivation of The Radial Equation

Derivation of The Radial Equation

The kinetic energy operator in spherical polar coordinates is

$\frac{\hat{p}^2}{2m_0} = -\frac{\hbar^2}{2m_0} \nabla^2 = - \frac{\hbar^2}{2m_0\,r^2}\left.$

The spherical harmonics satisfy

$\hat{l}^2 Y_{lm}(\theta,\phi)\equiv \left\{ -\frac{1}{\sin^2\theta} \left[ \sin\theta\frac{\partial}{\partial\theta} \Big(\sin\theta\frac{\partial}{\partial\theta}\Big) +\frac{\partial^2}{\partial \phi^2}\right]\right\} Y_{lm}(\theta,\phi) = l(l+1)Y_{lm}(\theta,\phi).$

Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,

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