Masujima M. Applied Mathematical Methods in Theoretical Physics
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2.5 Lippmann–Schwinger Equation 55
= (
b
,
a
) +
b
,
1
E
a
− H
0
+ iε
V
(+)
a
+
1
E
b
−E
a
− iε
(
b
, V
(+)
a
)
= δ
ba
+
1
E
a
− E
b
+ iε
+
1
E
b
−E
a
− iε
(
b
, V
(+)
a
)
= δ
ba
. (2.5.13)
Thus
(+)
a
forms a complete and orthonormal basis. The same proof goes through
for
(−)
a
also. Frequently, the orthonormality of
(±)
a
is assumed on the outset. We
have proven the orthonormality of
(±)
a
using the L–S equation and the formal
solution due to G. Chew and M. Goldberger.
In passing, we state that, in relativistic quantum field theory in the L.S.Z.
formalism, the outgoing wave
(+)
a
is called the in-state and is written as
(in)
a
,and
the incoming wave
(−)
a
is called the out-state and is written as
(out)
a
.
Unitarity of the
S
matrix:WedefinetheS matrix by
S
ba
= (
(−)
b
,
(+)
a
) = (
(out)
b
,
(in)
a
). (2.5.14)
This definition states that the S matrix transforms one complete set to the other
complete set. By making use of the formal solution of G. Chew and M. Goldberger
first and then using the L–S equation as before, we obtain
S
ba
= δ
ba
+
1
E
a
− E
b
+ iε
+
1
E
b
− E
a
+ iε
(
b
, V
(+)
a
) (2.5.1)
= δ
ba
− 2π iδ(E
b
−E
a
)(
b
, V
(+)
a
). (2.5.15)
We define the T matrix by
T
ba
= (
b
, V
(+)
a
). (2.5.16)
Then we have
S
ba
= δ
ba
− 2π iδ(E
b
−E
a
)T
ba
. (2.5.17)
If the S matrix is unitary, it satisfies
ˆ
S
†
ˆ
S =
ˆ
S
ˆ
S
†
= 1. (2.5.18)
These unitarity conditions are equivalent to the following conditions in terms of
the T matrix:
T
†
ba
− T
ba
=
2πi
n
T
†
bn
δ(E
b
−E
n
)T
na
,
2πi
n
T
bn
δ(E
b
−E
n
)T
†
na
,
with E
b
= E
a
. (2.5.19,20)
56 2 Integral Equations and Green’s Functions
In order to prove the unitarity of the S matrix, Eq. (2.5.18), it suffices to prove Eqs.
(2.5.19) and (2.5.20), which are expressed in terms of the T matrix.
We first note
T
†
ba
= T
∗
ab
= (
a
, V
(+)
b
)
∗
= (V
(+)
b
,
a
) = (
(+)
b
, V
a
).
Then
T
†
ba
−T
ba
= (
(+)
b
, V
a
) −(
b
, V
(+)
a
).
Inserting the formal solution of G. Chew and M. Goldberger to
(+)
a
and
(+)
b
above, we have
T
†
ba
−T
ba
= (
b
, V
a
) +
1
E
b
− H + iε
V
b
, V
a
−(
b
, V
a
) −
b
, V
1
E
a
− H + iε
V
a
= (V
b
,
1
E
b
− H − iε
−
1
E
b
− H + iε
V
a
)
= (V
b
,2π iδ(E
b
− H)V
a
), (2.5.21)
where, in the one line above the last line of Eq. (2.5.21), we used the fact that
E
b
= E
a
. Inserting the complete orthonormal basis,
(−)
, between the product of
the operators in Eq. (2.5.21), we have
T
†
ba
−T
ba
=
n
(V
b
,
(−)
n
)2πiδ(E
b
− E
n
)(
(−)
n
, V
a
)
= 2πi
n
T
†
bn
δ(E
b
− E
n
)T
na
.
This is Eq. (2.5.19).
If we insert the complete orthonormal basis,
(+)
, between the product of the
operators in Eq. (2.5.21), we obtain
T
†
ba
−T
ba
=
n
(V
b
,
(+)
n
)2πiδ(E
b
− E
n
)(
(+)
n
, V
a
)
= 2πi
n
T
bn
δ(E
b
− E
n
)T
†
na
.
This is Eq. (2.5.20). Thus the S matrix defined by Eq. (2.5.14) is unitary. The
unitarity of the S matrix is equivalent to the fact that the outgoing wave set
{
(+)
a
} and the incoming wave set {
(−)
b
} form the complete orthonormal basis,
respectively.
Actually, the S matrix has to be unitary since the S matrix transforms one
complete orthonormal set to another complete orthonormal set.
2.5 Lippmann–Schwinger Equation 57
Optical Theorem: The probability per unit time for the transition a → b is given by
w
ba
= 2πδ(E
b
− E
a
)
|
T
ba
|
2
. (2.5.22)
If we sum over the final state b,wehave
w
a
=
b
w
ba
= 2π
b
δ(E
b
− E
a
)
|
T
ba
|
2
. (2.5.23)
We compare the right-hand side of Eq. (2.5.23) with the special case of the unitarity
condition, Eq. (2.5.19),
T
†
aa
− T
aa
= 2π i
n
T
†
an
δ(E
a
−E
n
)T
na
.
We immediately obtain
w
a
=−2ImT
aa
. (2.5.24)
The cross section for a → (arbitrary state) is given by
σ
a
=
V
v
rel
w
a
, (2.5.25)
where V is the normalization volume and v
rel
is the relative velocity. From Eqs.
(2.5.24) and (2.5.25), we obtain
σ
a
=−
2V
v
rel
Im T
aa
. (2.5.26)
This relationship is called the optical theorem and holds true under any circum-
stance.Wenotethatthetotal cross section σ
a
includes the inelastic channels as well as
the elastic channel.
Asymptotic form: We shall consider the asymptotic form of the wavefunction
(+)
a
in the L–S equation. We construct the asymptotic form of
(+)
a
as the resulting
expression for the S matrix acting upon the free wavefunction
a
,
(+)
a
≈
ˆ
S
a
=
a
− 2π iδ(E
a
− H
0
)V
(+)
a
. (2.5.27)
We compare Eq. (2.5.27) with the L–S equation for
(+)
a
,
(+)
a
=
a
+
1
E
a
− H
0
+iε
V
(+)
a
.
We find that the asymptotic form of
(+)
a
is obtained by the replacement,
1
E
a
− H
0
+ iε
→−2πiδ(E
a
− H
0
).
58 2 Integral Equations and Green’s Functions
Equivalently, in closed form, we have
(+)
a
≈
a
−2πiδ(E
a
−H
0
)[(E
a
− H
0
)
(+)
a
], (2.5.28)
where (E
a
− H
0
)actson
(+)
a
and then δ(E
a
− H
0
)actson[(E
a
− H
0
)
(+)
a
].
We note the following:
(E
a
− H
0
)S
a
= 0. (2.5.29)
Namely, the asymptotic form of
(+)
a
satisfies the free particle equation. We can
say that the initial state and the final state described by the asymptotic form of
(±)
a
are the free particle state. In relativistic quantum field theory, the initial state
and the final state are the interacting system, respectively. Lehmann, Symanzik,
and Zimmermann (L.S.Z.) extended the notion of the asymptotic wavefunction
to the asymptotic condition in relativistic quantum field theory and successfully
constructed relativistic quantum field theory in the L.S.Z. formalism axiomatically,
with the notion of the in-state and the out-state.
The asymptotic condition, roughly speaking, is equivalent to the adiabatic
switching hypothesis of the interaction.
Rearrangement collision: So far, all the computations are formal. The L–S equation
shows its utmost power for the rearrangement collision where the splitting of H
into H
0
and V is not unique. The typical process is
n + d → n +n
+ p. (2.5.30)
In this case, the total Hamiltonian is
H = T
p
+ T
n
+ T
n
+ V
np
+V
n
p
+ V
nn
. (2.5.31)
Here T’s represent the kinetic energies and V’s the two-body potentials.
The decomposition of the total Hamiltonian H in the initial state is
H
initial
0
= T
p
+ T
n
+ T
n
+ V
n
p
,
V
initial
= V
np
+ V
nn
.
Here we included V
n
p
in H
0
in order to form the deuteron in the initial state. The
decomposition of the total Hamiltonian H in the final state is
H
final
0
= T
p
+ T
n
+ T
n
,
V
final
= V
np
+ V
nn
+ V
n
p
.
In order to discuss the reaction wherein the decomposition of the total Hamilto-
nian is different in the initial and final states, we introduce the two decompositions
H = H
a
+ V
a
initial state
= H
b
+ V
b
final state
, (2.5.32)
2.5 Lippmann–Schwinger Equation 59
where H
a
and H
b
are the free Hamiltonians in the initial and final states,
respectively. We introduce the free wavefunctions ,
a
and
b
,suchthat
(H
a
− E
a
)
a
= (H
b
− E
b
)
b
= 0. (2.5.33)
In order that the reaction a → b is possible, from the energy conservation, we must
have E
a
= E
b
.
We begin with the general collision problem starting from the initial state
wavefunction,
a
, and construct the asymptotic form of the final state wavefunction.
The formal solution to the L–S equation which satisfies the outgoing wave
condition, starting from
a
,is
(+)
a
=
a
+
1
E
a
− H + iε
V
a
a
. (2.5.34)
In order to construct the asymptotic wavefunction corresponding to the final
state, we use the operator identity
1
A
−
1
B
=
1
B
(B − A)
1
A
.
Then we have
1
E
a
− H + iε
−
1
E
a
− H
b
+ iε
=
1
E
a
−H
b
+ iε
V
b
1
E
a
− H + iε
. (2.5.35)
Substituting Eq. (2.5.35) into (2.5.34), we have
(+)
a
=
a
+
1
E
a
− H
b
+ iε
1 +V
b
1
E
a
− H + iε
V
a
a
. (2.5.36)
In the second term of Eq. (2.5.36), we now make the replacement for the asymptotic
wavefunction,
1
E
a
− H
b
+ iε
→−2πiδ(E
a
− H
b
)withE
a
= E
b
. (2.5.37)
The transition matrix element T
ba
is now expressed as
T
ba
=
b
,
1 +V
b
1
E
a
− H + iε
V
a
a
=
1 +
1
E
b
− H − iε
V
b
b
, V
a
a
= (
(−)
b
, V
a
a
). (2.5.38)
We have two ways of expressing T
ba
. Namely,
T
ba
= (
(−)
b
, V
a
a
) = (
b
, V
b
(+)
a
). (2.5.39)
60 2 Integral Equations and Green’s Functions
We prove the latter equality with the use of the formal solution of G. Chew and
M. Goldberger. By taking the difference of (
(−)
b
, V
a
a
)and(
b
, V
b
(+)
a
), on setting
E
a
= E
b
= E,weobtain
(
(−)
b
, V
a
a
) −(
b
, V
b
(+)
a
) = (
b
, V
a
a
) − (
b
, V
b
a
)
+
b
, V
b
1
E − H + iε
V
a
a
−
b
, V
b
1
E − H + iε
V
a
a
= (
b
,(V
a
− V
b
)
a
) = (
b
,(H
b
− H
a
)
a
)
= (E
b
−E
a
)(
b
,
a
) = 0.
We established Eq. (2.5.39). In the plane wave Born approximation,wereplace
(−)
b
and
(+)
a
with
b
and
a
, respectively, and we have
T
(Born)
ba
= (
b
, V
a
a
) = (
b
, V
b
a
).
Final state interaction: In the presence of the final state interaction, we split the
total Hamiltonian into the following form:
H = H
0
+ U + V, (2.5.40)
(1) U is strong and V is weak,
|
U
|
|
V
|
,
(2) we have the exact solution for H
0
+ U,
(H
0
+ U)χ
(±)
a
= E
a
χ
(±)
a
. (2.5.41)
For the electron in interaction with the electromagnetic field, for example, we have
H
0
= (free electron) +(free electromagnetic field),
U = (Coulomb potential),
V = (interaction between electron and electromagnetic field).
We represent the eigenstate of the total Hamiltonian H as
(±)
:
H
(±)
a
= E
a
(±)
a
. (2.5.42)
We have the following relationship between
(±)
a
and χ
(±)
a
:
(±)
a
= χ
(±)
a
+
1
E
a
− H ± iε
Vχ
(±)
a
= χ
(±)
a
+
1
E
a
− (H
0
+U) ± iε
V
(±)
a
. (2.5.43)
2.5 Lippmann–Schwinger Equation 61
This is the L–S equation relating
(±)
a
to χ
(±)
a
,withH
0
+U as the free Hamiltonian.
We can envision the photoelectric effect for the present treatment:
χ
(+)
a
: (1s electron) ×(1 photon state),
χ
(−)
b
: (electron in scattering state) ×(0 photon state),
with (χ
(−)
b
, χ
(+)
a
) = 0.
From the S matrix, S
ba
= (
(−)
b
,
(+)
a
), we pick up the first-order term in V,
(
(−)
b
,
(+)
a
) = (χ
(−)
b
, χ
(+)
a
) +
χ
(−)
b
,
1
E
a
− (H
0
+U) + iε
Vχ
(+)
a
+
1
E
b
− (H
0
+ U) − iε
Vχ
(−)
b
, χ
(+)
a
+···. (2.5.44)
In many applications, we frequently have (χ
(−)
b
, χ
(+)
a
) = 0 which we shall assume.
Keeping the first-order term in V in Eq. (2.5.44), we have
S
ba
=−2πiδ(E
a
−E
b
)(χ
(−)
b
, Vχ
(+)
a
). (2.5.45)
Hence we have the following T matrix:
T
ba
= (χ
(−)
b
, Vχ
(+)
a
). (2.5.46)
The relationship between
b
and χ
(−)
b
is given by
χ
(−)
b
=
b
+
1
E
b
−H
0
− iε
Uχ
(−)
b
=
b
+
1
E
b
−(H
0
+ U) − iε
U
b
. (2.5.47)
This is the L–S equation with H
0
+ U as the total Hamiltonian.
Thus the Born approximation to Eq. (2.5.46) is given by
T
(Born)
ba
= (
b
, Vχ
(+)
a
), (2.5.48)
and is frequently called as the distorted wave Born approximation.
We shall consider the following reaction:
p +p → p +n + π
+
.
We represent the potential energy between the two nucleons by U and the
interaction term necessary to produce π
+
by V. We assume
|
U
|
|
V
|
.The
wavefunction of the initial state χ
(+)
pp
satisfies the L–S equation,
χ
(+)
pp
=
pp
+
1
E
a
− H
a
+iε
U
pp
χ
(+)
pp
.
62 2 Integral Equations and Green’s Functions
Here, H
a
represents the kinetic energy of the two-nucleon system and U
pp
represents the potential energy between the two protons, corresponding to V
a
. E
a
represents the total energy of the initial state. The wavefunction of the final state
χ
(−)
pnπ
satisfies the L–S equation,
χ
(−)
pnπ
+
=
pnπ
+
+
1
E
a
− H
b
−iε
V
b
χ
(−)
pnπ
+
.
Here, H
b
represents the sum of the kinetic energies of p, n and π
+
,andtherest
mass of π
+
,andV
b
represents the three-body interaction term. We shall assume
that the interaction between π
+
and the nucleons is weak and we can approximate
V
b
by U
pn
. Then the wavefunction of the final state χ
(−)
pnπ
+
is separated as
χ
(−)
pnπ
+
≈ g
(−)
pn
h
π
+
.
The T matrix T
ba
is given by
T
ba
= (χ
(−)
b
, Vχ
(+)
a
) ≈ (g
(−)
pn
h
π
+
, Vχ
(+)
pp
).
The final proton–neutron wavefunction g
(−)
pn
satisfies the L–S equation,
g
(−)
pn
= g
(0)
pn
+
1
E
p
+ E
n
− H
p
−H
n
−iε
U
pn
g
(−)
pn
.
Here, H
p
and H
n
represent the kinetic energy operators of the proton and the
neutron, respectively, and E
p
and E
n
represent their eigenvalues. This is the
approach followed by K. Watson. We shall not go into any further computational
details.
Inclusion of the spin–spin force and the tensor force,
V(r) = V
0
(r) +V
spin-spin
(r)(σ
n
σ
p
) + V
tensor
(r)
3
(σ
n
r)(σ
p
r)
r
2
− (σ
n
σ
p
)
,
is immediate with the use of the spin-singlet and spin-triplet basis for the spin
wavefunction of the two-nucleon system.
For further details on rearrangement collision and final state interaction, we refer
the reader to the advanced textbooks on quantum mechanics and nuclear physics.
2.6
Scalar Field Interacting with Static Source
Consider the quantized real scalar field φ(
x, t) interacting with the time-independent
c-number source ρ(
x). Let the equation of motion for the field φ(
x, t)be
¨
φ(
x , t) −s
2
∇
2
φ(
x , t) +s
2
µ
2
φ(
x , t) = ρ(
x ), with µ = 0. (2.6.1)
2.6 Scalar Field Interacting with Static Source 63
The Hamiltonian of the system is given by
H =
1
2
d
x {
2
(
x , t) +s
2
(
∇φ(
x , t))
2
+s
2
µ
2
φ
2
(
x , t) −2ρ(
x )φ(
x , t)}. (2.6.2)
Imposing the equal-time canonical commutation relations,
[(
x , t), φ(
x
, t)] =−iδ(
x −
x
),
[(
x, t), (
x
, t)] = [φ(
x, t), φ(
x
, t)] = 0,
the Heisenberg equations of motion become
i
˙
φ(
x , t) = [φ(
x, t), H] = i(
x , t),
i
˙
(
x , t) = [(
x , t), H] = i{s
2
∇
2
φ(
x , t) −s
2
µ
2
φ(
x , t) +ρ(
x )},
(2.6.3a)
which agree with (2.6.1). We can write the q-number operators satisfying (2.6.3a)
in the large box of volume V in Schr
¨
odinger picture as
φ(
x,0)≡ φ(
x ) =
√
/V
k
(1/
2ω
k
){a
k
exp[i
k ·
x] + a
†
k
exp[−i
k ·
x ]},
(
x ,0)≡ (
x ) =−i
√
/V
k
ω
k
/2{a
k
exp[i
k ·
x ] − a
†
k
exp[−i
k ·
x]},
where
[a
k
, a
†
k
] = δ
k
k
,
[a
k
, a
k
] = [a
†
k
, a
†
k
] = 0,
ω
k
= s
k
2
+ µ
2
. (2.6.3b)
Substituting the expansions of φ(
x)and(
x) into (2.6.2), we obtain
H =
k
ω
k
a
†
k
a
k
+
1
2
+λ
k
a
k
+ λ
∗
k
a
†
k
.
Here, we set
ρ(
k) =
1
√
V
d
x ρ(
x )exp[i
k ·
x ], λ
k
=−
2ω
k
ρ(
k).
We can express the Hamiltonian as
H =
k
ω
k
a
†
k
+
λ
k
ω
k
a
k
+
λ
∗
k
ω
k
−
λ
k
2
ω
k
+
ω
k
2
. (2.6.4)
The last two terms above are the c-number terms which do not cause any
problem. The first term above represents the shift of a
k
and a
†
k
by the c-
numbers. Since the shifted a
k
and a
†
k
also satisfy the commutation relation
64 2 Integral Equations and Green’s Functions
(2.6.3b), we can express the shift as a result of the unitary transformation. If we
define
v ≡ exp
k
1
ω
k
!
λ
k
a
k
− λ
∗
k
a
†
k
"
,
we have
v
−1
a
k
v = a
k
−
λ
∗
k
ω
k
, v
−1
a
†
k
v = a
†
k
−
λ
k
ω
k
.
Here we used the Baker–Campbell–Hausdorff formula,
exp[−iB]A exp[iB] = A + i[A, B] +
i
2
2!
[[A, B], B] +···. (2.6.5)
Applying the unitary transformation v to the Hamiltonian H,weobtain
v
−1
H v =
k
ω
k
a
†
k
a
k
−
λ
k
2
ω
k
+
ω
k
2
≡ H
0
.
We can obtain the eigenvalue of H
0
immediately. We set the difference of H and
H
0
as
H
int
≡ H − H
0
=
k
λ
k
a
k
+ λ
∗
k
a
†
k
+
λ
k
2
ω
k
.
In order to understand the meaning of the Interaction Picture, using this H
0
,we
compute H
int
(t)definedby
H
int
(t) ≡ exp
i
H
0
t
H
int
exp
−i
H
0
t
.
We obtain, with the use of (2.6.5),
H
int
(t) =
k
λ
k
a
k
exp[−iω
k
t] +λ
∗
k
a
†
k
exp[iω
k
t] +
λ
k
2
ω
k
.
Connecting the state vector
(t)
%
in the Interaction Picture and the state vector
|
in the Heisenberg Picture by
(t)
%
= U(t)
|
,
we have
i
d
dt
U(t) = H
int
(t)U(t), (2.6.6)