2
1 Essentials of Non-relativistic Two-Body Scattering Theory
which is found to satisfy inhomogeneous integral equation. Finally, for a Hermitian
potential, we have the unitarity relation for the S-matrix which ensures conservation
of probability flux. This leads us to establish the well-known optical theorem relating
the imaginary part of the forward amplitude to the total cross section.
1.2 Boundary Condition of the Scattering State: Basic
Concepts
The scattering of two particles is described by the time-dependent Schrodinger
equation:
i
∂ψ
(+)
a (t)
∂t
= H ψ
(+)
a (t),
(1.1)
where H is the Hamiltonian which consists of (i) the kinetic energy, p
2
/(2μ) = H 0 ,
of the relative motion of the two particles plus (ii) the potential V, which is a function
of the relative coordinates of the particles. The scattering state ψ
(+)
a
develops from a
wave packet which is free in the infinite past and whose properties are characterized
by the index a. The solution of Eq. (1.1) is given by
ψ
(+)
a (t) = exp(−i Ht/) ψ
(+)
a
(1.2)
The boundary condition is formulated by introducing a reference wave packet φ a
at t = 0 and its time dependence is given by the free Hamiltonian H 0 :
φ a (t) = exp(−i H 0 t/)φ a
(1.3)
It is required that in the infinite past the wave function ψ
(+)
a
coincides with the
free packet φ a . In the limit of t → −∞, since the wave packets spread out in time,
it is not advisable to demand a pointwise agreement. That is why the strong limit
is needed [Note, however, in real experiment the wave packets associated with the
beam and detector are highly localized over experimental time scales.]. Thus, instead
of requiring a pointwise agreement, we demand that the norm of the difference be
zero, i.e.,
Lim
t→∞
exp(−i Ht/)ψ
(+)
a
− exp(−i H 0 t/)φ a
= 0
( 1 . 4 )
This limit on the norm, which is called a strong limit, is denoted by ‘s-lim,’ which
implies a certain restriction on the range of the potential. Equation (1.4) can thus be
expressed as:
ψ
(+)
a
= s− lim
t→−∞
exp(i Ht/) exp(−i H 0 t/)φ a ≡ Ω
(+)
φ a
(1.5)
1 Essentials of Non-relativistic Two-Body Scattering Theory
which is found to satisfy inhomogeneous integral equation. Finally, for a Hermitian
potential, we have the unitarity relation for the S-matrix which ensures conservation
of probability flux. This leads us to establish the well-known optical theorem relating
the imaginary part of the forward amplitude to the total cross section.
1.2 Boundary Condition of the Scattering State: Basic
Concepts
The scattering of two particles is described by the time-dependent Schrodinger
equation:
i
∂ψ
(+)
a (t)
∂t
= H ψ
(+)
a (t),
(1.1)
where H is the Hamiltonian which consists of (i) the kinetic energy, p
2
/(2μ) = H 0 ,
of the relative motion of the two particles plus (ii) the potential V, which is a function
of the relative coordinates of the particles. The scattering state ψ
(+)
a
develops from a
wave packet which is free in the infinite past and whose properties are characterized
by the index a. The solution of Eq. (1.1) is given by
ψ
(+)
a (t) = exp(−i Ht/) ψ
(+)
a
(1.2)
The boundary condition is formulated by introducing a reference wave packet φ a
at t = 0 and its time dependence is given by the free Hamiltonian H 0 :
φ a (t) = exp(−i H 0 t/)φ a
(1.3)
It is required that in the infinite past the wave function ψ
(+)
a
coincides with the
free packet φ a . In the limit of t → −∞, since the wave packets spread out in time,
it is not advisable to demand a pointwise agreement. That is why the strong limit
is needed [Note, however, in real experiment the wave packets associated with the
beam and detector are highly localized over experimental time scales.]. Thus, instead
of requiring a pointwise agreement, we demand that the norm of the difference be
zero, i.e.,
Lim
t→∞
exp(−i Ht/)ψ
(+)
a
− exp(−i H 0 t/)φ a
= 0
( 1 . 4 )
This limit on the norm, which is called a strong limit, is denoted by ‘s-lim,’ which
implies a certain restriction on the range of the potential. Equation (1.4) can thus be
expressed as:
ψ
(+)
a
= s− lim
t→−∞
exp(i Ht/) exp(−i H 0 t/)φ a ≡ Ω
(+)
φ a
(1.5)