Appendix A — Quick review for hydrogenic atoms

(This is a quick reference; more details are given in Section 21.3.)

We consider the solution for a “hydrogenic” atom, i.e. a bound state of a single electron in the electric field generated by a nucleus with total charge Z: V(r) = -\frac{Ze^2}{r}. Clearly to be a bound state, the electron energy must be less than zero. Since the Coulomb potential is a central potential (spherically symmetric), the solution to the Schrödinger equation splits into radial and angular components, \psi_{nlm}(r, \theta, \phi) = R_{nl}(r) Y_l^m(\theta, \phi). Solving the radial equation for the hydrogen atom is a difficult task, and as we’ve seen in other examples the solutions are written only in terms of special functions, either the associated Laguerre polynomials L_p^q(\rho) or the confluent hypergeometric functions, which are written as F(a;c;\rho). Here is the general formula in Sakurai’s notation:

R_{nl}(r) = \frac{1}{(2l+1)!} \left( \frac{2Zr}{na_0}\right)^l e^{-Zr / (na_0)} \\ \times \sqrt{ \left( \frac{2Z}{na_0} \right)^3 \frac{(n+l)!}{2n(n-l-1)!} } F\left(-n+l+1; 2l+2; \frac{2Zr}{na_0} \right)

and here it is again conveniently in Mathematica code:

R[n_, l_] := 1/(2*l + 1)!*((2*Z*r)/(n*a0))^l*Exp[(-Z*r)/(n*a0)]*
Sqrt[((2*Z)/(n*a0))^3*(n + l)!/(2 n*(n - l - 1)!)]*
Hypergeometric1F1[-n + l + 1, 2 l + 2, (2*Z*r)/(n*a0)];

The constant length scale a_0 is known as the Bohr radius, a_0 = \frac{\hbar^2}{m_e e^2}. As we observed generally for a particle in a central potential, the wavefunctions will be peaked away from the origin except for l=0; in particular only l=0 gives R_{n0}(0) \neq 0. The energy eigenstates are labelled by the principal quantum number, n, which is a linear combination of a radial quantum number q and the angular momentum quantum number l: n = q + l + 1. This imposes the restrictions that l \leq n-1 and n \geq 1; for the n=1 state, only l=0 is allowed. The energy eigenvalues are E_n = -\frac{1}{2} mc^2 \frac{Z^2 \alpha^2}{n^2} where \alpha is the fine-structure constant, \alpha = \frac{e^2}{\hbar c}. The value of the ground-state n=1 energy for the hydrogen atom (Z=1) is sometimes used as a unit of energy, known as the Rydberg: we see that \textrm{Ry} = \frac{me^4}{2\hbar^2} = \frac{e^2}{2a_0}, and so E_n = -\frac{Z^2}{n^2}\ \textrm{Ry}. I’ll emphasize that the energy eigenvalues of this solution depend only on n, not l or m. The E_n energy level is thus made up of \sum_l (2l+1) = n^2 degenerate states.

The ground state radial wavefunction is relatively simple, and we should have it handy for some of the examples we’re about to work: it is equal to R_{10}(r) = 2 \sqrt{\frac{Z^3}{a_0^3}} e^{-Zr/a_0}.