Principles of Quantum Mechanics · Problem 12.6.11

The Hydrogen Atom — R. Shankar

Physics — Quantum Mechanics · isotropic-oscillator-spherical

Problem

*(1) By combining Eqs. (12.6.48) and (12.6.49) derive the two-term recursion relation. Argue that

C_0 \neq 0

U

is to have the right properties near

y = 0

. Derive the quantization condition, Eq. (12.6.50). (2) Calculate the degeneracy and parity at each

n

and compare with Exercise 10.2.3, where the problem was solved in Cartesian coordinates. (3) Construct the normalized eigenfunction

\psi_{nlm}

for

n = 0

and

1

. Write them as linear combinations of the

n = 0

and

n = 1

eigenfunctions obtained in Cartesian coordinates.