The Lippmann-Schwinger equation (after Bernard Lippmann and Julian Schwinger ) is used in quantum mechanical perturbation theory and especially in scattering theory . It has the form of an integral equation for the wave function sought and is an alternative to the direct solution of the Schrödinger equation , whereby the boundary conditions are in the definition of the Green functions used .
ψ
{\ displaystyle \ psi}
In quantum mechanical perturbation theory
In general, in perturbation theory, the Hamilton operator is broken down into the "free Hamilton operator" , for which a solution is known, and a part treated as a small perturbation ( potential ) :
H
{\ displaystyle H}
H
0
{\ displaystyle H_ {0}}
V
{\ displaystyle V}
H
=
H
0
+
V
{\ displaystyle H = H_ {0} + V}
Eigenfunctions of the free Hamilton operator satisfy the equation
|
ϕ
0
⟩
{\ displaystyle | \ phi _ {0} \ rangle}
(
E.
-
H
0
)
|
ϕ
0
⟩
=
0
{\ displaystyle \ left (E-H_ {0} \ right) | \ phi _ {0} \ rangle = 0}
where is the associated eigenvalue .
E.
{\ displaystyle E}
An operator is called a "free Green function " for which the following applies:
G
0
{\ displaystyle G_ {0}}
G
0
(
E.
-
H
0
)
|
ϕ
0
⟩
=
|
ϕ
0
⟩
{\ displaystyle G_ {0} \ left (E-H_ {0} \ right) | \ phi _ {0} \ rangle = | \ phi _ {0} \ rangle}
This operator is, so to speak, an inverse function of the free Hamilton operator . A mathematically correct representation requires the consideration of as distribution .
G
0
{\ displaystyle G_ {0}}
The unknown eigenfunctions of the complete Hamilton operator and its Green function are now defined in an analogous manner .
|
ψ
⟩
{\ displaystyle | \ psi \ rangle}
G
{\ displaystyle G}
The Lippmann-Schwinger equation thus applies:
|
ψ
⟩
=
|
ϕ
0
⟩
+
G
0
V
|
ψ
⟩
{\ displaystyle | \ psi \ rangle = | \ phi _ {0} \ rangle + G_ {0} V | \ psi \ rangle}
This equation is usually solved iteratively , with the restriction to the first non- trivial order being called Born's approximation .
In scattering theory
Accordingly, the Lippmann-Schwinger equation is mainly used in scattering theory. Here it is calculated how the wave function of a particle changes when scattering at a potential V, using the kinetic component for a free particle as the free Hamilton operator :
H
^
0
=
p
^
2
2
m
{\ displaystyle {\ hat {H}} _ {0} = {\ frac {{\ hat {\ mathbf {p}}} ^ {2}} {2m}}}
with the momentum operator .
p
^
{\ displaystyle {\ hat {\ mathbf {p}}}}
To derive the Lippmann-Schwinger equation for a stationary scattering problem, one starts from the Schrödinger equation:
[
-
ℏ
2
2
m
Δ
+
V
(
r
→
)
]
ψ
k
=
E.
k
ψ
k
{\ displaystyle \ left [- {\ frac {\ hbar ^ {2}} {2m}} \ Delta + V ({\ vec {r}}) \ right] \ psi _ {k} = E_ {k} \ psi _ {k}}
With
the kinetic energy of a free particle
E.
k
=
ℏ
2
k
→
2
2
m
{\ displaystyle E_ {k} = {\ frac {\ hbar ^ {2} {\ vec {k}} ^ {2}} {2m}}}
its direction of entry
k
→
k
{\ displaystyle {\ frac {\ vec {k}} {k}}}
its direction of spread
k
′
→
k
=
r
→
r
=
n
→
{\ displaystyle {\ frac {\ vec {k ^ {\ prime}}} {k}} = {\ frac {\ vec {r}} {r}} = {\ vec {n}}}
it should be noted that it is an elastic scattering, i. H. the magnitude of the momentum vector is not changed: and for all vectors .
k
′
=
k
{\ displaystyle k ^ {\ prime} = k}
|
v
→
|
=
v
{\ displaystyle | {\ vec {v}} | = v}
Converted and with the requirement results:
E.
≥
0
{\ displaystyle E \ geq 0}
[
Δ
+
k
2
]
ψ
k
=
2
m
ℏ
2
V
(
r
→
)
ψ
k
=:
U
(
r
→
)
ψ
k
{\ displaystyle \ left [\ Delta + k ^ {2} \ right] \ psi _ {k} = {\ frac {2m} {\ hbar ^ {2}}} V ({\ vec {r}}) \ psi _ {k} =: U ({\ vec {r}}) \ psi _ {k}}
This can be solved with the method of Green's functions:
[
Δ
+
k
2
]
ϕ
0
=
0
⇒
ϕ
0
=
1
(
2
π
)
3
/
2
e
i
k
→
r
→
[
Δ
+
k
2
]
G
(
r
→
)
=
δ
(
r
→
)
⇒
G
(
r
→
)
=
-
1
4th
π
e
i
k
r
r
ψ
k
(
r
→
)
=
ϕ
0
+
∫
d
3
r
→
′
G
(
r
→
-
r
→
′
)
⋅
U
(
r
→
′
)
ψ
k
(
r
→
′
)
{\ displaystyle {\ begin {aligned} \ left [\ Delta + k ^ {2} \ right] \ phi _ {0} = 0 \ quad & \ Rightarrow \ quad \ phi _ {0} = {\ frac {1 } {(2 \ pi) ^ {3/2}}} e ^ {i {\ vec {k}} {\ vec {r}}} \\\ left [\ Delta + k ^ {2} \ right] G ({\ vec {r}}) = \ delta {({\ vec {r}})} \ quad & \ Rightarrow \ quad G ({\ vec {r}}) = - {\ frac {1} { 4 \ pi}} {\ frac {e ^ {ikr}} {r}} \\\ psi _ {k} ({\ vec {r}}) = \ phi _ {0} & + \ int d ^ { 3} {\ vec {r}} ^ {\ prime} G ({\ vec {r}} - {\ vec {r}} ^ {\ prime}) \ cdot U ({\ vec {r}} ^ { \ prime}) \ psi _ {k} ({\ vec {r}} ^ {\ prime}) \\\ end {aligned}}}
This results in the Lippmann-Schwinger equation of the scattering theory:
ψ
k
(
r
→
)
=
1
(
2
π
)
3
/
2
e
i
k
→
⋅
r
→
-
2
m
4th
π
ℏ
2
∫
d
3
r
→
′
⋅
e
i
k
|
r
→
-
r
→
′
|
|
r
→
-
r
→
′
|
V
(
r
→
′
)
ψ
k
(
r
→
′
)
{\ displaystyle \ psi _ {k} ({\ vec {r}}) = {\ frac {1} {(2 \ pi) ^ {3/2}}} e ^ {i {\ vec {k}} \ cdot {\ vec {r}}} - {\ frac {2m} {4 \ pi \ hbar ^ {2}}} \ int d ^ {3} {\ vec {r}} ^ {\ prime} \ cdot {\ frac {e ^ {ik | {\ vec {r}} - {\ vec {r}} ^ {\ prime} |}} {| {\ vec {r}} - {\ vec {r}} ^ {\ prime} |}} V ({\ vec {r}} ^ {\ prime}) \ psi _ {k} ({\ vec {r}} ^ {\ prime})}
The location representation was explicitly chosen here.
This equation can be solved iteratively by replacing on the right-hand side with the solution obtained up to that point and choosing as the starting value of the iteration:
ψ
k
(
r
→
)
{\ displaystyle \ psi _ {k} ({\ vec {r}})}
ψ
k
(
0
)
(
r
→
)
=
ϕ
0
(
r
→
)
{\ displaystyle \ psi _ {k} ^ {(0)} ({\ vec {r}}) = \ phi _ {0} ({\ vec {r}})}
The first iteration
ψ
k
(
1
)
(
r
→
)
=
1
(
2
π
)
3
/
2
e
i
k
→
r
→
-
2
m
4th
π
ℏ
2
∫
d
3
r
→
′
⋅
e
i
k
|
r
→
-
r
→
′
|
|
r
→
-
r
→
′
|
V
(
r
→
′
)
1
(
2
π
)
3
/
2
e
i
k
→
r
→
′
{\ displaystyle \ psi _ {k} ^ {(1)} ({\ vec {r}}) = {\ frac {1} {(2 \ pi) ^ {3/2}}} e ^ {i { \ vec {k}} {\ vec {r}}} - {\ frac {2m} {4 \ pi \ hbar ^ {2}}} \ int d ^ {3} {\ vec {r}} ^ {\ prime} \ cdot {\ frac {e ^ {ik | {\ vec {r}} - {\ vec {r}} ^ {\ prime} |}} {| {\ vec {r}} - {\ vec { r}} ^ {\ prime} |}} V ({\ vec {r}} ^ {\ prime}) {\ frac {1} {(2 \ pi) ^ {3/2}}} e ^ {i {\ vec {k}} {\ vec {r}} ^ {\ prime}}}
is then the already mentioned Born approximation in position representation.
Individual evidence
^ Bernard Lippmann and Julian Schwinger : Variational principles for scattering processes. I . In: Physical Review . tape 79 , no. 3 , 1950, p. 469-480 , doi : 10.1103 / PhysRev.79.469 . Equation 1.84 on p. 475.
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