Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
(Hint: Reduce the given equation to Bessel's equation ®rst.)
7.20 The lengthening simple pendulum: Consider a small mass m suspended by a
string of length l. If its length is increased at a steady rate r as it swings back
and forth freely in a vertical plane, ®nd the equation of motion and the
solution for small oscillations.
7.21 Evaluate the integrals:
Z
…a† x n J n1 …x†dx;
Z
…b†
x n J n‡1 …x†dx;
Z
…c†
x 1 J 1 …x†dx:
7.22 In quantum mechanics, the three-dimensional SchroÈ dinger equation is
@ …r; t†
ip ˆ p2
p
@t 2m r2 …r; t†‡V …r; t†; i ˆ
1 ; p ˆ h=2:
(a) When the potential V is independent of time, we can write …r; t† ˆ
u…r†T…t†. Show that in this case the SchroÈ dinger equation reduces to
p2
2m r2 u…r†‡Vu…r† ˆEu…r†;
a time-independent equation along with T…t† ˆe iEt=p , where E is a
separation constant.
(b) Show that, in spherical coordinates, the time-independent SchroÈ dinger
equation takes the form
p2
2m
"
1 @
r 2 r 2 @u
‡ 1
#
@ @u 1 @ 2 u
@r @r r 2 sin ‡
sin @ @ r 2 sin 2 @
‡ V…r†u ˆ Eu;
then use separation of variables, u…r;;†ˆR…r†Y…; †, to split it into
two equations, with as a new separation constant:
p2 1 d
2m r 2 r 2 dR
‡ V ‡
dr dr r 2 R ˆ ER;
p2 1 @ @Y
sin
p2 1 @ 2 Y
2m sin @ @ 2m sin 2 @ 2 ˆ Y:
It is straightforward to see that the radial equation is in the form of Eq.
(7.99). Continuing the separation process by putting Y…; † ˆ…†…†,
the angular equation can be separated further into two equations, with
as separation constant:
p2 1 d 2
2m d 2 ˆ ;
p2
2m sin d d
d
sin
d
sin 2 ‡ ˆ 0:
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