1.3 Moller Operator
3
1.3 Moller Operator
The right-hand side of Eq. (1.5) introduces a symbol
(+) , which operating on a free
state produces the scattering state ψ
(+)
a . This defines the Moller operator
(+) , which
is explicitly written as:
(±)
= s− lim
t→∓∞
exp(i Ht/) exp(−i H 0 t)
(1.6)
Clearly, the Moller operator
(−) operating on the free state produces the scattering
state ψ
(−)
a (t), which approaches the free state φ a (t) for t → +∞. From the definition
(1.6), it follows that for any finite τ
s− lim
t→∓∞
exp(i H(t + τ/) exp(−i H 0 (t + τ )/)
=
(±)
(τ ) = exp(i Hτ/))
(±) exp(−i Hτ/)
(1.7)
is also valid. By differentiating with respect to τ , we get
d
(±)
dτ
= exp(i Hτ )
H
(±)
−
(±) H 0
exp(−i H 0 τ ) = 0,
which gives the commutation relation
H Ω
(±)
= Ω
(±) H 0
(1.8)
(Note from here onwards we are using the units for which = 1).
The domain of Moller operator is the Hilbert space of square-integrable free
states. However, work by Faddeev [7] has shown that this domain can be extended to
include states of sharp momentum. We are thus allowed to use momentum eigenstates
|
p instead of the wave packets φ a and apply the operators
(±) on them to get
scattering states |
p
(±) . Thus, for instance, the operator
(±) operating on the state |
p
gives the scattering state |
p
(+) consisting of a plane wave exp(ikz) which describes
the incoming flux and the flux of the unscattered particles plus a spherical wave
f (θ, ϕ) exp(ikr)/r which describes the flux of the scattered particles. The scattering
states |
p
(±) have a sharp energy E just as the free states |
p have the sharp energy
p
2
/2μ. The two energies are the same, as can be shown by using the commutation
relation (1.8):
H |
p
(±) = H Ω
(±)
|
p =
(±) H 0 |
p
=
(±) p
2
2μ
|
p =
p
2
2μ
|
p
(±)
(1.9)
However, applying the Hermitian conjugate operator
(±)
∗ on the bound states
ψ n of H, we find
3
1.3 Moller Operator
The right-hand side of Eq. (1.5) introduces a symbol
(+) , which operating on a free
state produces the scattering state ψ
(+)
a . This defines the Moller operator
(+) , which
is explicitly written as:
(±)
= s− lim
t→∓∞
exp(i Ht/) exp(−i H 0 t)
(1.6)
Clearly, the Moller operator
(−) operating on the free state produces the scattering
state ψ
(−)
a (t), which approaches the free state φ a (t) for t → +∞. From the definition
(1.6), it follows that for any finite τ
s− lim
t→∓∞
exp(i H(t + τ/) exp(−i H 0 (t + τ )/)
=
(±)
(τ ) = exp(i Hτ/))
(±) exp(−i Hτ/)
(1.7)
is also valid. By differentiating with respect to τ , we get
d
(±)
dτ
= exp(i Hτ )
H
(±)
−
(±) H 0
exp(−i H 0 τ ) = 0,
which gives the commutation relation
H Ω
(±)
= Ω
(±) H 0
(1.8)
(Note from here onwards we are using the units for which = 1).
The domain of Moller operator is the Hilbert space of square-integrable free
states. However, work by Faddeev [7] has shown that this domain can be extended to
include states of sharp momentum. We are thus allowed to use momentum eigenstates
|
p instead of the wave packets φ a and apply the operators
(±) on them to get
scattering states |
p
(±) . Thus, for instance, the operator
(±) operating on the state |
p
gives the scattering state |
p
(+) consisting of a plane wave exp(ikz) which describes
the incoming flux and the flux of the unscattered particles plus a spherical wave
f (θ, ϕ) exp(ikr)/r which describes the flux of the scattered particles. The scattering
states |
p
(±) have a sharp energy E just as the free states |
p have the sharp energy
p
2
/2μ. The two energies are the same, as can be shown by using the commutation
relation (1.8):
H |
p
(±) = H Ω
(±)
|
p =
(±) H 0 |
p
=
(±) p
2
2μ
|
p =
p
2
2μ
|
p
(±)
(1.9)
However, applying the Hermitian conjugate operator
(±)
∗ on the bound states
ψ n of H, we find