2.1 The Schr ̈odinger equation 27can be cast in a convenient form by the substitutionP(r)=rR(r):
−
^2
2 med^2 P
dr^2+
{
^2
2 mel(l+1)
r^2−
e^2 / 4 π 0
r−E
}
P=0. (2.16)
The term proportional tol(l+1)/r^2 is the kinetic energy associated
with the angular degrees of freedom; it appears in this radial equation
as an effective potential that tends to keep wavefunctions withl=0
away from the origin. Dividing through this equation byE =−|E|
(a negative quantity sinceE0 for a bound state) and making the
substitution
ρ^2 =2 me|E|r^2
^2(2.17)
reduces the equation to the dimensionless form
d^2 P
dρ^2+
{
−
l(l+1)
ρ^2+
λ
ρ− 1
}
P=0. (2.18)
The constant that characterises the Coulomb interaction strength is
λ=e^2
4 π 0√
2 me
^2 |E|. (2.19)
The standard method of solving such differential equations is to look for
a solution in the form of a series. The series solutions have a finite num-
ber of terms and do not diverge whenλ=2n,wherenis an integer.^1515 The solution has the general form
P(ρ)=Ce−ρv(ρ), wherev(ρ)isan-
other function of the radial coordinate,
for which there is a polynomial solution
(see Woodgate 1980 and Rae 1992).
Thus, from eqn 2.19, these wavefunctions have eigenenergies given by^16
(^16) Using eqn 1.41.
E=−
2 me(
e^2 / 4 π 0) 2
^2
1
λ^2=−hcR∞1
n^2. (2.20)
This shows that the Schr ̈odinger equation has stationary solutions at en-
ergies given by the Bohr formula. The energy does not depend onl;this
accidentaldegeneracy of wavefunctions with differentlis a special fea-
ture of Coulomb potential. In contrast, degeneracy with respect to the
magnetic quantum numbermlarises because of the system’s symmetry,
i.e. an atom’s properties are independent of its orientation in space, in
the absence of external fields.^17 The solution of the Schr ̈odinger equation^17 This is true for any spherically-
gives much more information than just the energies; from the wavefunc- symmetric potentialV(r).
tions we can calculate other atomic properties in ways that were not
possible in the Bohr–Sommerfeld theory.
We have not gone through the gory details of the series solution, but
we should examine a few examples of radial wavefunctions (see Ta-
ble 2.2). Although the energy depends only onn, the shape of the
wavefunctions depends on bothnandland these two quantum num-
bers are used to label the radial functionsRn,l(r). Forn=1thereis
only thel= 0 solution, namelyR 1 , 0 ∝e−ρ.Forn=2theorbital
angular momentum quantum number isl= 0 or 1, giving
R 2 , 0 ∝(1−ρ)e−ρ,
R 2 , 1 ∝ρe−ρ.