64
2 The Quantum Approach to the Two-Body Problem
In the high-energy limit (i.e., when kd >> 1) the asymptotic expressions of the
Bessel and Hankel functions provide us with additional insight
tan δ l
kd>>1
∼ −
sin(kd − lπ/2)
cos(kd − lπ/2)
(2.93)
where δ l = − arctan(kd − lπ/2). The fact that kd >> 1 implies that many partial
waves contribute to the total cross section. For the calculation of this, we note that
the sum over l in Eq. (2.79) can be replaced by an integral
24
l max
l=0
l max
0
(2l + 1)ds l
2
max (kd)
2
.
(2.94)
In addition, the factor sin
2
δ l in the summation that appears in the definition of the
total cross section can be approximated by the average value 1/2 (approximation of
the random phase), such that we will simply note
25
σ tot =
4π
k 2
1
2
(kd)
2
= 2πd
2
.
(2.95)
The fact that for high energies we obtain a quantum cross section which is twice the
classical one is surprising because for kd >> 1 (or equivalently for the De Broglie
wavelength 2πd >> λ) you would expect to find correspondence with the classical
limit. The origin of this discrepancy is a result of using a discontinuous potential
in r = d doing that the scattering cannot be described classically. The “extra” πd
2
factor derives from the interference between the incident and outgoing wave at small
scattering angles where one cannot distinguish the incident and outgoing waves. In
other words, at high energies, a contribution from diffractive wave-like nature of the
quantum system adds up to the classical mechanics contribution.
2.3.3 The Morse Potential
In the previous chapter, we have discussed the validity of the Morse potential and
its use for the classical treatment of atom–atom scattering. To the end of developing
24 The considered approximation l max kr stems from the fact that the effective potential due to
the rotation can be considered of the same order of the incident energy E, namely:
l(l + 1) 2
2mr 2 E or l max
√
2mE
r kr
25 The logical procedure adopted for obtaining this result is an example of a heuristic procedure.
2 The Quantum Approach to the Two-Body Problem
In the high-energy limit (i.e., when kd >> 1) the asymptotic expressions of the
Bessel and Hankel functions provide us with additional insight
tan δ l
kd>>1
∼ −
sin(kd − lπ/2)
cos(kd − lπ/2)
(2.93)
where δ l = − arctan(kd − lπ/2). The fact that kd >> 1 implies that many partial
waves contribute to the total cross section. For the calculation of this, we note that
the sum over l in Eq. (2.79) can be replaced by an integral
24
l max
l=0
l max
0
(2l + 1)ds l
2
max (kd)
2
.
(2.94)
In addition, the factor sin
2
δ l in the summation that appears in the definition of the
total cross section can be approximated by the average value 1/2 (approximation of
the random phase), such that we will simply note
25
σ tot =
4π
k 2
1
2
(kd)
2
= 2πd
2
.
(2.95)
The fact that for high energies we obtain a quantum cross section which is twice the
classical one is surprising because for kd >> 1 (or equivalently for the De Broglie
wavelength 2πd >> λ) you would expect to find correspondence with the classical
limit. The origin of this discrepancy is a result of using a discontinuous potential
in r = d doing that the scattering cannot be described classically. The “extra” πd
2
factor derives from the interference between the incident and outgoing wave at small
scattering angles where one cannot distinguish the incident and outgoing waves. In
other words, at high energies, a contribution from diffractive wave-like nature of the
quantum system adds up to the classical mechanics contribution.
2.3.3 The Morse Potential
In the previous chapter, we have discussed the validity of the Morse potential and
its use for the classical treatment of atom–atom scattering. To the end of developing
24 The considered approximation l max kr stems from the fact that the effective potential due to
the rotation can be considered of the same order of the incident energy E, namely:
l(l + 1) 2
2mr 2 E or l max
√
2mE
r kr
25 The logical procedure adopted for obtaining this result is an example of a heuristic procedure.