13.7 Bessel functions 385
The associated Laguerre polynomials arise in the solution of the radial part of the
Schrödinger equation for the hydrogen atom (Section 14.6), and they occur there in
the form of associated Laguerre functions
(13.44)
forn 1 = 1 0, 1, 2, 3, =,l 1 = 1 0, 1, 2, =, (n 1 − 11 ). These functions satisfy the differential
equation
(13.45)
and they are orthogonal with respect to the weight functionx
2in the interval
01 ≤ 1 x 1 ≤ 1 ∞:
(13.46)
13.7 Bessel functions
The Bessel equation is
5x
2y′′ 1 + 1 xy′ 1 + 1 (x
21 − 1 n
2)y 1 = 10 (13.47)
where nis a real number. This equation ranks with the Legendre equation in its
importance in the physical sciences, although it is met less frequently in chemistry
than in physics and engineering. Bessel functions are involved, for example, in the
solution of the classical wave equation for the vibrations of circular and spherical
membranes, and the same solutions are found for the Schrödinger equation for the
particle in a circular box and in a spherical box. The functions are important in the
formulation of the theory of scattering processes.
Equation (13.47), when divided by x
2, is of type (13.3) and is solved by the
Frobenius method; that is, by expressing the solution in the form (13.4)
y(x) 1 = 1 x
r(a
01 + 1 a
1x 1 + 1 a
2x
21 +1-)
=
−−!
,′2
1
3nnl
nl
nn()
()
δ
ZZ
020221∞∞fxf xxdx exL x
nl n lxlnll, ′−++() () = ()
,LLxxdx
nll′+21 2+()
′′+ ′+−
−
f =
x
f
n
x
ll
x
f
211
4
0
2()
fxexL x
nlxlnll,−/++()= ()
2215‘At this junction
It is time to wrestle
With a well-known function
Due to Herr Bessel’.
Friedrich Wilhelm Bessel (1784–1864), German astronomer. Examples of Bessel functions were discussed by
Daniel Bernoulli, Euler, and Lagrange, but the first systematic study appeared in 1824 in a paper by Bessel on
perturbations of planetary orbits. Bessel is known as the recipient of numerous letters from his close friend Gauss.
In 1810 Gauss wrote: ‘This winter I am giving two courses of lectures to three students, of whom one is only
moderately prepared, the other is less than moderately, and the third lacks both preparation and ability. Such are
the burdens =’