3.2 Problems
139
(a) Put F(r ) = exp (−r/ν) y(r ), where E = −1/(2ν
2 ), and show that
d
2 y
dx 2 =
2
ν
d
dr
−
ν
r
y
(b) Assuming that y(r ) can be expanded as the series
y(r ) =
∞
p=0
a p r
p+1
,
Show that the coefficients a p in the series satisfy the recurrence relation,
p( p + 1)a p =
2
ν
( p − ν)a p−1
(c) Solutions of the radial Schrodinger equation exist which are bounded for
all r provided that ν = n, where n is a positive integer. Show that the
un-normalized radial function for the n = 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
=
< p x >
m
(b)
d < p x >
dt
=< −∂V/∂x >
3.15 Consider the time-independent Schrodinger equation in three dimensions
−
2
2m
∇
2
+ V (r )
ψ = Eψ
In spherical coordinates
∇
2
=
1
r 2
∂
∂r
r
2 ∂
∂r
+
1
r 2 sin θ
∂
∂θ
sin θ
∂
∂θ
+
1
r 2 sin
2
θ
∂
2
∂ϕ 2
(a) Write ψ(r, θ, ϕ) = ψ r (r )Y (θ, ϕ) as a separable solution and split
Schrodinger’s equation into two independent differential equations, one
depending on r and the other depending on θ and ϕ.
(b) Further separate the angular equation into θ and ϕ parts
(c) Combine the angular part and the potential part of the radial equation and
write them as an effective potential V e . Then make the substitution χ(r ) =
r ψ r (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 angular equation. The angular equation is given by
−
1
sin θ
∂
∂θ
sin θ
∂
∂θ
+
1
sin
2
θ
∂
2
∂ϕ 2
Y (θ, ϕ) = λY (θ, ϕ)
Solve the equation and, in the process, derive the quantum number m.