References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 8.14.
The general wave function can be written as the product of a radial function and a spherical harmonic .
With the substitution the radial equation for hydrogen can be written
The effective potential is the term in parentheses:
and has the form of a well with sloping sides on both sides, at points and defined by the roots of
In this case, the WKB equations satisfy the condition
Plugging in the formulas for the hydrogen atom we get
where we’ve factored out (since for a bound state, ). We can simplify the notation by defining the positive quantities
The turning points and can be found from the roots of the term inside the square root in 6:
so we can factor the quadratic to get
since the ground state of hydrogen is given by the Bohr formula