Masujima M. Applied Mathematical Methods in Theoretical Physics
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558 10 Calculus of Variations: Applications
(e) Next we consider the case,
w(x) = r
2
, Q =
01
r
2
0
.
Assuming that Q(x) is constant in the interval [x, x + a], show that S(x + a, x)
can be expressed as
S(x + a, x) =
r
−12
0
0 r
12
exp
01
10
ar
r
12
0
0 r
−12
=
cosh ar,(1r)sinhar
r sinh ar,coshar
.
By regarding r as the continuous and differentiable function r(x)ofx,and
assuming that
r
2r
is small compared with
|
r
|
,weobtain
S(x, x
0
) =
(r
0
r)
12
cosh F (rr
0
)
−12
sinh F
(rr
0
)
12
sinh F (rr
0
)
12
cosh F
with F =
x
x
0
rdx.
(f) Near the turning point, say x = a where s(a) = r(a) = 0 with the assumption
that E > V(x)forx < a (classically allowed zone) and E < V(x)forx > a
(classically forbidden zone), show that the wavefunction ψ(x) can be written
generally as
ψ(x) =
C
1
√
s
exp
i
x
a
sdx
+
C
2
√
s
exp
−
i
x
a
sdx
, x < a,(E)
ψ(x) =
C
3
√
r
exp
−
1
x
a
rdx
+
C
4
√
r
exp
1
x
a
rdx
, x > a. (F)
The wavefunction should be dumped in the classically forbidden zone,
E < V(x)(x > a). Show that we must choose C
4
= 0 in Eq. (F),
ψ(x) =
C
2
√
r
exp
−
1
x
a
rdx
, x > a. (G)
(g) Near the turning point where the WKB approximation is not valid, we
assume that
E − V(x) ≈ F
0
(x − a), F
0
=−
dV(x)
dx
x=a
< 0.
10.20 Problems for Chapter 10 559
In a region where Eq. (G) is valid, show that we have
ψ(x) =
C
2
√
2m
|
F
0
|
1
(x − a)
14
exp
−
1
x
a
9
2m
|
F
0
|
(x − a)dx
. (H)
We first let x pass x = a from the classically forbidden zone (x > a)tothe
classically allowed zone (x < a) along a semicircle of radius ρ in the upper
half plane. On this semicircle, show that we have
x − a = ρ exp[iθ ], θ :0→ π ,
x
a
9
(x − a)dx =
2
3
ρ
32
cos
3
2
θ + i sin
3
2
θ
.
At the end of the semicircle, we have the pure imaginary exponent,
−
i
x
a
9
2m
|
F
0
|
(a − x)dx =−
i
x
a
sdx .
The change of the prefactor of the exponential in Eq. (H) is given by
(x − a)
−14
→ (a − x)
−14
exp
%
−i
π
4
'
.
Show that the connection formula for C
2
is given by
C
2
=
C
2
exp
%
−i
π
4
'
.
Likewise, the connection formula for C
1
is obtained by letting x pass x = a
from the classically forbidden zone (x > a) to the classically allowed zone
(x < a) along a semicircle of radius ρ in the lower half plane. Show that
C
1
=
C
2
exp
%
i
π
4
'
.
Show that the wavefunction in the forbidden zone (x > a) corresponds to the
wavefunction in the allowed zone (x < a),
ψ(x) =
C
√
s
cos
1
x
a
sdx +
π
4
.
Show also that, independent of the number of and the location of the turning
points, the connection formula from the classically forbidden zone to the
classically allowed zone is given by
C
2
√
r
exp
−
1
x
a
rdx
for the region V(x)>E
→
C
√
s
cos
1
x
a
sdx
−
π
4
for the region V(x)<E
.
560 10 Calculus of Variations: Applications
Hint: We cite the following article and the books for the discussion of WKB
approximation.
Klein, O.: Z. Physik. 80, (1932), 792.
Bender, Carl M., and Orszag, Steven A.: Advanced Mathematical Methods For
Scientists and Engineers, Springer, New York, 1999.
Pauli, W.: General Principles of Quantum Mechanics, Springer, Heidelberg, 1980;
Helv. Phys. Acta. 5, 179, (1932).
10.41. (a) Show that the Lagrangian density appropriate for the Laplace equation
is
L = (∇ψ)
2
.
(b) Show that, for two conductors with a fixed potential difference V
0
,the
electrostatic energy is given by (1/2)CV
2
0
,whereC is the capacity.
(c) Show that the variational principle for the capacity is
δ[C] = δ
1/4πV
2
0
(
∇ψ
)
2
dV
= 0.
10.42. Consider the scalar Helmholtz equation
∇
2
ψ + k
2
ψ = 0,
where ψ satisfies either homogeneous Dirichlet or homogeneous Neumann conditions
on a surface S. Show that the variational principle for k
2
is
δ
)
k
2
*
= δ
−
ψ∇
2
ψdV/
ψ
2
dV
.
Upon application of Green’s theorem, show that the variational principle for k
2
becomes
δ
)
k
2
*
= δ
(∇ψ)
2
dV/
ψ
2
dV
.
10.43. Consider the vibration of a clamped circular membrane. Let the radius of
the membrane be a. The boundary conditions for ψ are ψ(a) = 0andtheoverall
requirement that ψ be continuous in value and gradient inside the boundary.
Consider only the circularly symmetric modes so that ψ is a function of r only.
Then, show that the variational principle is
δ
)
k
2
*
= δ
a
0
(dψ/dr)
2
rdr/
a
0
ψ
2
rdr
.
Introduce the dimensionless independent variable x = r/a.Then
10.20 Problems for Chapter 10 561
δ
)
(ka)
2
*
= δ
1
0
(dψ/dx)
2
xdx/
1
0
ψ
2
xdx
.
Choose the possible trial functions as (1 −x
2
), which vanishes at x = 1andatthe
same time has a continuous gradient at x = 0. Obtain
)
(ka)
2
*
= 6.
10.44. Consider the reaction matrix, satisfying the integral equation
R = V + VG
0
0
R. (A)
A related wavefunction ψ
a
may be defined by the integral equation
ψ
a
= χ
a
+ G
0
0
Vψ
a
,(B)
where χ
a
is an unperturbed wavefunction satisfying Kχ
a
= ε
a
χ
a
.
(a) Show that a matrix element of R is written as
R
ba
= (χ
b
, Rχ
a
) = (χ
b
, Vψ
a
) = (ψ
b
, Vχ
a
), (C)
where ψ
b
also satisfies (B) with χ
a
replaced with χ
b
. Show that we can express
R
ba
as
R
ba
=
ψ
b
,(V − VG
0
0
V)ψ
a
. (D)
Show that Eqs. (C) and (D) permit us to set
R
ba
=
(ψ
b
, Vχ
a
)(χ
b
, Vψ
a
)
(ψ
b
,(V − VG
0
0
V)ψ
a
)
. (E)
Show that it is stationary with respect to small variations of either ψ
a
or ψ
b
about its correct form:
(
δR
ba
|
δψ
a
= 0,
δR
ba
|
δψ
b
= 0.
(F)
(b) Consider the generalization of this result to the T-matrix:
T = V + VG
+
V = V + VG
+
0
T,(G)
where the wavefunctions satisfy
ψ
±
a
= χ
a
+G
±
0
Vψ
±
a
. (H)
562 10 Calculus of Variations: Applications
Show that
T
ba
= (χ
b
, Tχ
a
) = (χ
b
, Vψ
+
a
) = (ψ
−
b
, Vχ
a
) =
(ψ
−
b
, Vχ
a
)(χ
b
, Vψ
+
a
)
(ψ
−
b
,(V − VG
+
0
V)ψ
+
a
)
.
(I)
Show that, if we perform the variation ψ
+
a
→ ψ
+
a
+ δψ
+
a
or
ψ
−
b
→ ψ
−
b
+ δψ
−
b
, respectively, we have to the first order in the variation,
(
δT
ba
|
δψ
+
a
= 0,
δT
ba
|
δψ
−
b
= 0.
(J)
(c) As an illustration, let us replace ψ
+
a
and ψ
−
b
by the trial functions χ
a
and χ
b
,
respectively, in the expression (I) for T
ba
. If we expand this in power of V,
show that
T
ba
= (χ
b
, Vχ
a
) +(χ
b
, VG
+
0
Vχ
a
) +···. (K)
This agrees with the Born series,
T = V + VG
+
0
V + VG
+
0
VG
+
0
V +···=T
(1)
+T
(2)
+···,
through the second order.
(d) In application of (I), consider the case of simple potential scattering with
V = V(r) and the choice of trial functions,
(
ψ
+
a
= (2π)
−3/2
exp[iκ ·
r],
ψ
−
b
= (2π )
−3/2
exp
)
iκ
f
·
r
*
.
Show that
T(κ
f
·κ, κ
f
, κ) = (2π)
−3
exp
)
−i(κ
f
−κ) ·
r
*
V(r)d
3
r
2
×
exp
)
−i(κ
f
−κ) ·
r
*
V(r)d
3
r
+
d
3
rd
3
r
exp[−iκ
f
·
r
]V(r
)
2M
r
exp
)
iκ
r −
r
*
4π
r −
r
V(r)
exp[iκ ·
r]
−1
. (L)
10.20 Problems for Chapter 10 563
(e) Show that Eq. (E) provides also a variational principle for the partial wave
scattering amplitudes. One needs only identify R
l
with −(2/π)κ tan δ
l
, V with
the reduced potential v,andG
0
0
with G
0l
. Then, on setting ψ
a
= ψ
b
= w
(0)
l
,
show that
−
2
π
κ tan δ
l
=
(y
l
, vw
(0)
l
)(w
(0)
l
, vy
l
)
(w
(0)
l
,(v − vG
0l
v)w
(0)
l
)
. (M)
Show that the expression (M) is stationary with respect to small variations of
w
(0)
l
about its correct value.
Hint for Problem 10.44: We cite the following textbook for variational principle
due to Schwinger for phase shift and variational principle due to Kohn for atomic
scattering.
Goldberger, M.L. and Watson, K.M.: Collision Theory, John Wiley & Sons, New
York, 1964, Chapters 6 and 11.
10.45. Consider the case of a particle in a central potential and look for a stationary
expression for the coefficient T
l
of the expansion of the transition amplitude in
spherical harmonics,
T
b←a
= 16π
2
lm
(2l + 1)T
l
Y
∗
lm
(
ˆ
k
b
)Y
lm
(
ˆ
k
a
). (A)
(a) Show that, if ψ
(+)
a
is the complete stationary scattering wave, the partial wave
ψ
l
satisfies the integral equation
ψ
l
(r) = j
l
(kr) +G
(+)
l
Vψ
l
(r), (B)
where G
(+)
l
is the integral operator whose kernel is Green’s function,
G
(+)
l
(r, r
) ≡−(2mk/
2
)j
l
(kr
<
)h
(+)
l
(kr
>
).
In other words,
G
(+)
l
Vψ
l
(r) ≡
∞
0
G
(+)
l
(r, r
)V(r
)ψ
l
(r
)r
2
dr
.
Let
ϕ
1
, ϕ
2
denote the scalar product of two radial functions ϕ
1
, ϕ
2
:
ϕ
1
, ϕ
2
≡
∞
0
ϕ
∗
1
(r)ϕ
2
(r)r
2
dr.
The integral form for T
l
can therefore be written as
T
l
=
j
l
, Vψ
l
.
564 10 Calculus of Variations: Applications
(b) Let us now put
A[ψ] ≡
j
l
, Vψ
=
ψ
∗
, Vj
l
, B[ψ] ≡
ψ
∗
,(V − VG
(+)
l
V)ψ
.
Consider the functional,
T
l
[ψ] ≡
A
2
B
.
Show that
T
l
[ψ
l
] = T
l
.
(c) Show that T
l
takes the same value T
l
for any function ψ
l
satisfying the
following less restrictive condition than (B):
ψ
l
(r) = Cj
l
(kr) +G
(+)
l
Vψ
l
(r)ifV(r) = 0; C, arbitrary constant. (C)
Calculating the variation of T
l
as a function of δψ, show that
δT
l
=
2A
B
δA −
A
2
B
2
δB,
δA[ψ] =
δψ
∗
, Vj
l
,
δB[ψ] = 2
δψ
∗
,(V − VG
(+)
l
V)ψ
.
Write
δT
l
=
2A
B
2
δψ
∗
, F
,
where F(r)isgivenby
F(r) = B[ψ]V(r)j
l
(kr) −A[ψ]V(r)(ψ − G
(+)
l
Vψ).
In order to have δT
l
= 0foranyδψ, it is necessary and sufficient that F(r)
vanish. For this, it is necessary that j
l
(kr)andψ − G
(+)
l
Vψ be proportional at
any point where V(r)isnotzero,thatis,thatψ be one of the functions ψ
l
obeying equation (C). Thus, by adjusting the value of C in Eq. (C), we obtain
T
l
= T
l
|
st
. (D)
10.20 Problems for Chapter 10 565
(d) Separate the real and imaginary parts of G
(+)
l
(r, r
). The real part is Green’s
function,
G
(1)
l
(r, r
) ≡−(2mk/
2
)j
l
(kr
<
)n
l
(kr
>
).
and
G
(+)
l
(r, r
) = G
(1)
l
(r, r
) −i(2mk/
2
)j
l
(kr)j
l
(kr
).
Substituting this expression in the definition of T
l
, show that
T
−1
l
= T
l
−1
st
=
ψ
∗
,
V − VG
(1)
l
V
ψ
j
l
, Vψ
l
2
st
+ i
2mk
2
.
From
T
l
=−
2
exp[iδ
l
]sinδ
l
/2mk,
show that
T
−1
l
−i(2mk/
2
) =−2mk cot δ
l
/
2
.
Obtain the following stationary expression for k cot δ
l
:
k cot δ
l
=−
2
2m
ψ
∗
,
V − VG
(1)
l
V
ψ
j
l
, Vψ
l
2
st
. (E)
(e) Substituting the free wave j
l
(kr)forψ in the right-hand side of (E), show that
k cot δ
l
=−
2
2m
1 −
l
j
l
, Vj
l
with
l
≡
j
l
, VG
(1)
l
Vj
l
j
l
, Vj
l
. (F)
Consider the S-wave scattering by a square well in the low energy limit. Take
V(r) =
(
−V
0
, r < r
0
,
0, r > r
0
.
Calculate the scattering length,
a =−lim
k→0
(k cot δ)
−1
,
566 10 Calculus of Variations: Applications
by the different methods and compare the results with that given by the exact
calculation. Setting
b =
2mV
0
r
2
0
/
2
1/2
,
show that
exact calculation:
a =−
tan b
b
− 1
r
0
.
variational calculation [formula (F)]:
a
var
=
(1/3)b
2
1 −(2/5)b
2
r
0
.
first-order Born approximation:
a
(1)
B
=−
1
3
b
2
r
0
.
second-order Born approximation:
a
(2)
B
=−
1
3
b
2
+
2
15
b
4
r
0
.
567
References
Local Analysis and Global Analysis
We cite the following book for lo-
cal analysis and global analysis of
ordinary differential equations.
1. Bender, Carl M., and Orszag, Steven
A.: Advanced Mathematical Methods
For Scientists And Engineers, Springer,
New York, 1999.
Integral Equations
We cite the following book for the
theory of Green’s functions and
boundary value problems.
2. Stakgold, I.: Green’s Functions and
Boundary Value Problems, John Wiley
& Sons, New York, 1979.
We cite the following books for
general discussions of the theory of
integral equations.
3. Tricomi, F.G.: Integral Equations,
Dover, New York, 1985.
4. Pipkin, A.C.: A Course on Integral
Equations, Springer, New York, 1991.
5. Bach, M.: Analysis, Numerics and
Applications of Differential and Integral
Equations, Addison Wesley, Reading,
MA, 1996.
6. Wazwaz, A.M.: A First Course in
Integral Equations, World Scientific,
Singapore, 1997.
7. Polianin, A.D.: Handbook of Integral
Equations, C RC Press, Florida, 1998.
8. Jerri, A.J.: Introduction to Integral
Equations with Applications,2ndedi-
tion, John Wiley & Sons, New York,
1999.
We cite the following books for ap-
plications of integral equations to the
scattering problem in nonrelativistic
quantum mechanics, namely, the
Lippmann–Schwinger equation.
9. Goldberger, M.L., and Watson, K.M.:
Collision Theory, John Wiley & Sons,
New York, 1964,Chapter5.
10. Sakurai, J.J.: Modern Quantum Me-
chanics, Addison-Wesley, Reading,
MA, 1994,Chapter7.
11. Nishijima, K.: Relativistic Quan-
tum Mechanics, Baifuu-kan, Tokyo,
1973, Chapter 4, Section 4.11. (In
Japanese).
We cite the following book for appli-
cations of integral equations to the
theory of elasticity.
12. Mikhlin, S.G. et al: The Integral
Equations of the Theory of Elasticity,
Teubner, Stuttgart, 1995.
We cite the following book for ap-
plication of integral equations to
microwave engineering.
13. Collin, R.E.: Field Theory of Guided
Waves, O xford University Press,
Oxford, 1996.
We cite the following article for
application of integral equations to
chemical engineering.
14. Bazant, M.Z., and Trout, B.L.: Phys-
ica, A300, 139, (2001).
We cite the following book for phys-
ical details of dispersion relations in
classical electrodynamics.
15. Jackson, J.D.: Classical Electrodynam-
ics, 3rd edition, John Wiley & Sons,
New York, 1999, Section 7.10, p. 333.
We cite the following books for
applications of the Cauchy-type
integral equations to dispersion re-
lations in the potential scattering
Applied Mathematical Methods in Theoretical Physics, Second Edition. Michio Masujima
Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-40936-5