1000 Solved Problems in Modern Physics

3.2 Problems 139

(a) PutF(r)=exp (−r/ν)y(r), whereE=− 1 /(2ν^2 ), and show that

d^2 y

dx^2

=

2

ν

d

dr

−

ν

r

y

(b) Assuming thaty(r) can be expanded as the series

y(r)=

∑∞

p= 0

aprp+^1 ,

Show that the coefficientsapin the series satisfy the recurrence relation,

p(p+1)ap=

(

2

ν

)

(p−ν)ap− 1

(c) Solutions of the radial Schrodinger equation exist which are bounded for

allrprovided thatν =n, wherenis a positive integer. Show that the

un-normalized radial function for then=2 state is

F(r)=a 0 e−r/^2 r(1−r/2)

3.14 State Ehrenfest’s theorem. Show that

(a)

d<x>

dt

=

<px>

m

(b)

d<px>

dt

=<−∂V/∂x>

3.15 Consider the time-independent Schrodinger equation in three dimensions
[(
−

^2

2 m

)

∇^2 +V(r)

]

ψ=Eψ

In spherical coordinates

∇^2 =

(

1

r^2

)

∂

∂r

(

r^2

∂

∂r

)

+

(

1

r^2 sinθ

)

∂

∂θ

(

sinθ

∂

∂θ

)

+

(

1

r^2 sin^2 θ

)

∂^2

∂φ^2

(a) Write ψ(r,θ,φ) = ψr(r)Y(θ,φ) as a separable solution and split

Schrodinger’s equation into two independent differential equations, one

depending onrand the other depending onθandφ.

(b) Further separate the angular equation intoθandφparts

(c) Combine the angular part and the potential part of the radial equation and

write them as an effective potentialVe. Then make the substitutionχ(r)=

rψr(r) and transform the radial equation into a form that resembles the

one-dimensional Schrodinger equation.

3.16 Consider a three-dimensional spherically symmetrical system. In this case,
Schrodinger’s equation can be decomposed into a radial equation and an angu-
lar equation. The angular equation is given by

−

[(

1

sinθ

)(

∂

∂θ

)(

sinθ

∂

∂θ

)

+

(

1

sin^2 θ

)(

∂^2

∂φ^2

)]

Y(θ,φ)=λY(θ,φ)

Solve the equation and, in the process, derive the quantum numberm.