Masujima M. Applied Mathematical Methods in Theoretical Physics
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2.7 Problems for Chapter 2 75
with the normalization constant c
l
given by
c
l
=
i
l
+
2mk
π
.
By taking r very large, obtain the scattering amplitude for the lth partial
wave as
f
l
(k) = exp[iδ
l
]
sin δ
l
k
=−
2m
2
∞
0
j
l
(kr)A
l
(k;r)V(r)r
2
dr.
Hint for Problem 2.18: The wavefunction for the free wave in the spherical
polar coordinate is given by
*
x
Elm
%
= c
l
j
l
(kr)Y
l,m
(
ˆ
r).
When r →∞, use the asymptotic formula for j
l
(kr)andh
(1)
l
(kr).
2.19. Consider a scattering of nonrelativistic particle off delta-shell potential. The
potential we consider is
U(r) =−λδ(r − a), (A)
i.e., a force field that vanishes everywhere except on a sphere of radius a.
The strength parameter λ has the dimension (length)
−1
. One can look upon
(A) as a crude model of the interaction experienced by a neutron when it
interacts with a nucleus of radius a.
(a) Show that the radial integral equation
A
l
(k;r) = j
l
(kr) +
∞
0
G
(l)
k
(r;r
)U(r
)A
l
(k;r
)r
2
dr
,
with
G
(l)
k
(r;r
) =−ikj
l
(kr
<
)h
l
(kr
>
),
reduces to the algebraic equation,
A
l
(k;r) = j
l
(kr) +ikλa
2
A
l
(k;a) ×
j
l
(kr)h
l
(ka), r < a,
j
l
(ka)h
l
(kr), r > a.
(B)
Obtain, by setting r → a in (B),
A
l
(k;a) =
j
l
(ka)
1 −ikλa
2
j
l
(ka)h
l
(ka)
.
76 2 Integral Equations and Green’s Functions
(b) Construct the partial wave scattering amplitude from the general
formula
exp[iδ
l
]sinδ
l
=−k
∞
0
j
l
(kr)U(r)A
l
(k;r)r
2
dr,
as
exp[iδ
l
]sinδ
l
= kλa
2
j
l
(ka)A
l
(k;a) =
kλa
2
[j
l
(ka)]
2
1 −ikλa
2
j
l
(ka)h
l
(ka)
. (C)
The tangent of δ
l
is also a convenient quantity for some purposes.
Using h
l
(z) = j
l
(z) + in
l
(z), we obtain
tan δ
l
=
kλa
2
[j
l
(ka)]
2
1 +kλa
2
j
l
(ka)n
l
(ka)
.
It is natural to express all length in units of a, and all wave numbers in
units of 1/a. Hence define the dimensionless variables as
ρ = r/a, ξ = ka, g = λa , in terms of which, show that (C) is expressed
as
exp[iδ
l
]sinδ
l
=
gξ[j
l
(ξ)]
2
1 −iξ gj
l
(ξ)h
l
(ξ)
. (D)
(c) The existence of bound states requires the occurrence of poles in the
radial continuum wavefunctions when the latter are treated as
functions of the complex variable k.Hencewemustdeterminethe
location of the pure imaginary zeros of the function
D
l
(g;ζ ) = 1 −iζ gj
l
(ζ )h
l
(ζ ),
where ζ = ξ + iη. We call the iterative solution of the radial integral
equation the Born series for A
l
, and we call the corresponding
expansion,
1
k
exp[iδ
l
]sinδ
l
=−
∞
0
[j
l
(kr)]
2
U(r)r
2
dr
−
∞
0
j
l
(kr)U(r)G
(l)
k
(r, r
)
U(r
)j
l
(kr
)r
2
r
2
drdrr
+···,
the Born series for the partial wave amplitude. Show that
exp[iδ
l
]sinδ
l
= gξ[j
l
(ξ)]
2
∞
n=0
g
n
[iξj
l
(ξ)h
l
(ξ)]
n
,
2.7 Problems for Chapter 2 77
and its radius of convergence in the complex g-plane is
g
<
1
ξ
j
l
(ξ)h
l
(ξ)
≡ g
(B)
l
(ξ).
(d) Show that the actual form of the radial wavefunction for a bound state
can be obtained by evaluating the residue of (B) at the bound state
pole.Ifthispoleisatζ = iη
(l)
b
,say,showthatwithN the normalization
constant,
R
l
(ρ) =
Nj
l
(iη
(l)
b
ρ)h
l
(iη
(l)
b
), ρ<1,
Nh
l
(iη
(l)
b
ρ)j
l
(iη
(l)
b
), ρ>1.
(e) Consider the behavior of the S-wave cross section as a function of
energy; this is given by sin
2
δ
0
(ξ). From (D), show that
sin
2
δ
0
(ξ) =
g
2
sin
4
ξ
(ξ − (1/2)g sin 2ξ )
2
+ g
2
sin
4
ξ
. (E)
This function attains its maximum value of unity at the roots of
2ξ = g sin 2ξ. (F)
Show that the resonances fall into two very different classes, ξ
(n)
b
and
ξ
(n)
s
:(a)ξ
(n)
b
near ξ = π/2, 3π/2, 5π/2 ...,whicharebroad because
sin
4
ξ ≈ 1 at these points, and (b) ξ
(n)
s
near ξ = π,2π,3π, ...,which
are very sharp because sin
4
ξ ≈ 0 there. Verify these statements. The
location of the sharp resonances follows immediately from (F);
that is,
ξ
(n)
s
nπ
1 +
1
g
= Re ζ
n
if g n.
Furthermore, sin
2
ξ
(n)
s
(nπ/g)
2
. Show that we can write (E) in the
Breit–Wigner form
sin
2
δ
0
(ξ)
|
Im ζ
n
|
2
(ξ − Re ζ
n
)
2
+
|
Im ζ
n
|
2
,(G)
in the immediate vicinity of the sharp resonances. Show that as n
increases, the resonances become broader, and the whole discussion
culminating in (G) are only valid if the width
|
Im ζ
n
|
is small compared
to the distance between neighboring resonances, i.e.,
|
Im ζ
n
|
Re ζ
n+1
−Re ζ
n
. (H)
78 2 Integral Equations and Green’s Functions
Once (H) is violated, the resonances soon disappear.
Hint for Problem 2.19: For the clean-cut general discussion of the
delta-shell potential problem along the line of the present problem, as
well as the analytic properties of physical quantities of our interest as a
function of k and energy, we refer the reader to the following textbook.
Gottfried, K.: Quantum Mechanics, Vol. I : Fundamentals,W.A.
Benjamin, New York. (1966). Chapter III, Section 15.
2.20. Consider the S-wave scattering off a spherically symmetric potential U(r).
The radial Schr
¨
odinger equation is given by
d
2
dr
2
u(r) + k
2
u(r) = U(r)u(r),
with the boundary conditions
u(0) = 0andu(r) ∼
sin(kr + δ)
sin δ
as r →∞.
(a) Consider the rigid sphere potential problem,
U(r) =
∞ for r < b,
0forr ≥ b.
Obtain the exact solution for the rigid sphere potential problem.
(b) Consider the spherical box potential problem,
U(r) =
U(> 0) for r < b,
0forr ≥ b.
For the spherical box potential problem, show that δ is given by
k cot δ =
(K cot Kb)(k cot kb) +k
2
k cot kb − K cot Kb
,
from the continuity of the wavefunction at r = b,whereK is given by
K =
k
2
− U.
(c) Consider the bound state for the spherical box potential problem,
U(r) =
U(< 0) for r < b,
0forr ≥ b.
2.7 Problems for Chapter 2 79
From the continuity of the wavefunction at r = b, show that
γ (γ
2
=−k
2
> 0) is determined by
K cot Kb =−γ ,
K =
k
2
− U =
|
U
|
−γ
2
.
2.21. Consider the S-wave scattering off a spherically symmetric potential U(r).
The radial Schr
¨
odinger equation is given by
d
2
dr
2
u(r) + k
2
u(r) = U(r)u(r),
with the boundary conditions
u(0) = 0andu(r) ∼
sin(kr + δ)
sin δ
as r →∞.
The scattering length a and the effective range r
eff
are defined by
k cot δ =−
1
a
+
1
2
r
eff
k
2
+···.
Potential of range b has a bound state of energy
E =−E
B
=−κ
2
B
< 0,
(i) Spherical box potential:
U(r) =
U(< 0) for r < b,
0forr ≥ b.
(a) Set up the equations for binding energy E =−E
B
and phase shift
δ (E > 0). Weakly bound means that
Kb >
π
2
, K =
k
2
− U,
just above π/2, but not much. Define an energy by
b
(
|
U
|
−) ≡
π
2
,
and q by
= q
2
.
Then
|
U
|
.
80 2 Integral Equations and Green’s Functions
(b) Show that the binding energy E
B
is related to by
κ
B
b
1
2
(qb)
2
.
(c) For positive E but E
|
U
|
,defineχ by
χ ≡−Kb cot(Kb).
Show that
χ
1
2
(k
2
+ q
2
)b
2
1
2
(kb)
2
+κ
B
b.
(d) From this, show that the scattering length a is given by
a ≡−lim
k→0
tan δ
k
b
1 +κ
B
b
κ
B
b
b.
(ii) δ shell potential:
U(r) =−κ
0
δ(r − b).
(e) Set up the equations for binding energy E =−E
B
and phase shift
δ (E > 0).
(f) Show that the bound state energy satisfies
1 +κ
B
b κ
0
b.
(g) Show that in this case too, the scattering length a is given by
a ≡−lim
k→0
tan δ
k
b
1 + κ
B
b
κ
B
b
.
This is the same result as for the spherical box potential of the
same range and the same binding energy.
(h) You are in a position to show, without any effort, that if is
negative, that is, no bound state exists, the scattering length has an
opposite sign, that is negative.
This problem is not too terribly intellectual, but it is a good
exercise in approximation techniques.
2.22. Consider a time-dependent wave packet. A solution of the time-dependent
Schr
¨
odinger equation,
2.7 Problems for Chapter 2 81
i
∂
∂t
− H
|
ψ
= 0,
can be constructed by superposition of energy eigenfunctions as
*
r
ψ
(+)
(t)
%
=
d
3
k
,
r
k
(+)
-
exp[−
i
E
k
t]a(
k −
k
0
),
with
E
k
=
(
k)
2
2m
,
where a(
k −
k
0
) is an amplitude containing a narrow band of
k-values only.
It is suggested that you make a packet in the z-direction (along
k
0
)only,
a(
k −
k
0
) = C exp
−
1
2
D
2
(k
z
−k
0
)
2
δ(k
x
)δ(k
y
).
D is the spatial width of the packet in the z-direction.
A narrow packet means that we assume
1
D
= k k
0
,
and
k · r
0
=
r
0
D
1,
where r
0
is the range the potential V(r)(thatisV(r) ≈ 0forr > r
0
). In
addition, we make a dynamic assumption that k
0
is not the position of a
narrow resonance. This allows us to assume that, for r r
0
,
*
r
k
(+)
%
∼
=
,
r
k
(+)
0
-
or V(r)
*
r
k
(+)
%
∼
=
V(r)
,
r
k
(+)
0
-
.
Analogously, with all the assumptions made above, we can construct an
incident wave packet as
*
r
φ(t)
%
=
d
3
k
,
r
k
-
exp[−
i
E
k
t]a(
k −
k
0
).
(a) Show that
*
r
φ(t)
%
∼
=
exp
ik
0
z −
i
E
0
t
F(z − v
0
t), E
0
=
(
k
0
)
2
2m
.
Establish the form of F(z − v
0
t). Show that it is a packet of width D.
82 2 Integral Equations and Green’s Functions
(b) Show that
*
r
ψ(t)
%
∼
=
*
r
φ(t)
%
+ f (θ )
exp[ik
0
r −
i
E
0
t]
r
F(
r
− v
0
t),
where f (θ ) is the scattering amplitude.
(c) Show that the scattering occurs only in the time interval
|
t
|
<
D
v
0
.
(d) Show that prior to the scattering,
t < −D/v
0
,
we have
*
r
ψ(t)
%
=
*
r
φ(t)
%
,
and that the scattered wave is present only after the scattering,
t > D/v
0
.
Hint for Problem 2.22: We cite the following article for the description
of the time-dependent scattering problem.
Low, F.E.: Brandeis Summer School Lectures. (1959).
2.23. Correct the error in the following proof:
T
ba
= (
b
, V
(+)
a
) = (
b
,(H − H
0
)
(+)
a
) = (E
a
− E
b
)(
b
,
(+)
a
).
Hence
T
ba
= 0whenE
a
= E
b
.
Hint: Use the L–S equation for
(+)
a
in the last term of the above expression.
2.24. Consider the general properties of the S matrix. The S matrix
,
k
S
k
-
=
,
k
(−)
k
(+)
-
=
d
3
r
,
k
(−)
r
-,
r
k
(+)
-
for the scattering of a spinless particle by a central force potential has the
following properties:
2.7 Problems for Chapter 2 83
(i) it is unitary,
S
†
S = SS
†
= 1,
(ii) it is diagonal in the energy,
S ∼ δ(E
k
−E
k
),
(iii) it is a scalar, that is, a function of
k
2
,
k
2
,and
k ·
k
only.
(a) Show that the general form of a matrix satisfying (i), (ii), and (iii) is
,
k
S
k
-
=
δ(k
− k)
4πk
k
l
(2l + 1)P
l
(
.
k
·
.
k)exp[2iδ
l
(k)].
For the proof, use the fact that the simplest unitary matrix
satisfying (i), (ii), and (iii) is the unit matrix,
,
k
k
-
= δ(
k
−
k) =
δ(k
− k)
4πk
k
l
(2l + 1)P
l
(
.
k
·
.
k)
=
δ(k
− k)
k
k
l,m
Y
lm
(
.
k
)Y
∗
lm
(
.
k).
(b) Show that S is a symmetric matrix,
,
k
S
k
-
=
,
k
S
k
-
.
For the proof of this, you need to consider the rotational
invariance and the time reversal relation,
,
r
k
(+)
-
=
,
r
−
k
(−)
-
∗
≡
,
−
k
(−)
r
-
.
2.25. Consider the scattering cross section for spin-1/2 particle. In this problem,
we obtain the differential cross section σ(θ ) and total cross section σ
tot
of
spin-1/2 particle off the spin-dependent potential of the form
V(r) = V
0
(r) +V
LS
(r)(
L ·σ).
(a) We first construct the scattering amplitude
*
m
s
f (θ )
m
s
%
,where
m
s
%
is
the eigenstate of σ
op
and P
l
(cos θ ) is the eigenfunction of (
L
op
)
2
,
(
L
op
)
2
P
l
(cos θ ) = l(l +1)P
l
(cos θ ).
84 2 Integral Equations and Green’s Functions
Out of P
l
(cos θ )
m
s
%
which is the eigenstate of (
L
op
)
2
and σ
op
, we select
the eigenkets of (
L ·σ ) for which the potential is diagonal,
J ≡
L +
1
2
σ
J
2
=
L
2
+
3
4
+ (
L ·σ ).
When
J
2
and
L
2
are diagonal, so is (
L ·σ ),
j = l +
1
2
:(
L ·σ )
l, j = l +
1
2
%
= l
l, j = l +
1
2
%
,
j = l −
1
2
:(
L ·σ )
l, j = l −
1
2
%
=−(l + 1)
l, j = l −
1
2
%
.
Show that the projection operators P
l,j
that project the state of the
definite j and definite l out of P
l
(cos θ )
m
s
%
are given by
P
l,j=l+1/2
=
l + 1 +(
L ·σ )
2l + 1
, P
l,j=l−1/2
=
l − (
L ·σ )
2l + 1
,
which satisfy the property of the projection operator,
P
l,j
P
l,j
= δ
jj
P
l,j
,
j
P
l,j
= 1.
(b) For the complete set of the basic kets P
l,j
P
l
(cos θ )
m
s
%
,(
L ·σ )is
diagonal now. The matrix element of the S matrix is given by
,
k
, m
s
S
k, m
s
-
=
δ(k
− k)
4πk
k
l
(2l + 1)
j
exp[2iδ
l,j
]
,
m
s
P
l,j
P
l
(cos θ )
m
s
-
.
By definition of the T matrix and the scattering amplitude f (θ ), we have
,
k
, m
s
(S − 1)
k, m
s
-
=−2πiδ(E
k
− E
k
)
,
k
, m
s
T
k, m
s
-
= i
δ(E
k
− E
k
)
2πk
,
k
, m
s
f (θ )
k, m
s
-
.
Show that the scattering amplitude f (θ)isgivenby
,
k
, m
s
f (θ )
k, m
s
-
=
1
k
l
(2l + 1)
j
exp[2iδ
l,j
] −1
2i
,
m
s
P
l,j
P
l
(cos θ )
m
s
-
=
1
k
l
(2l + 1)
j
exp[iδ
l,j
]sinδ
l,j
,
m
s
P
l,j
P
l
(cos θ )
m
s
-
.