time independent Schr\"odinger equation in spherical coordinates
When writing the time independent Schrödinger equation in spherical coordinates, we need to plug the Laplacian in Spherical Coordinates
into the time independent Schrödinger equation. The Laplacian
was found to be
Using the three dimensional Schrödinger equation we then have
We can gain insight into this somewhat ugly equation by rewriting it using the square of the angular momentum
operator in spherical polar coordinates:
This leads to
This equation is only exactly solvable if
, a function without angular dependence. We then write
leading to the following equation:
To solve this equation we need to remove the angular dependence. This is simply done by substituting the eigenfunctions of
into the equation. These are known to be the spherical harmonics,
. We also know that these have eigenvalues
, i.e.
We now substitute this result into the Schrödinger equation and divide through by a common factor of
This is the radial equation.
References
[1] Griffiths, D. "Introduction to Quantum Mechanics" Prentice Hall, New Jersey, 1995.
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