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3 Quantum Mechanics – II
R/λ
l=0
(2l + 1) = (R/ -
λ)
2
∴ σ r = σ s = π -
λ
2
R
-
λ 2
= π R
2
The total cross-section
σ t = σ r + σ s = 2π R
2
which is twice the geometrical cross-section
3.111 The potential which an electron sees as it approaches an atom of a monatomic
gas can be qualitatively represented by a square well. Slow particles are considered.
V (r ) = −V 0 ; r ≤ R
= 0;
r > R
corresponding to an attractive potential. Scattering of slow particles for
which kR << 1, is determined by the equation
∇
2
+ k
2
−
2μV
2
ψ 2 = 0 (inside the well)
(1)
with k
2
= 2μE/
2 , and the wave number k = p/
Outside the well the equation is
(∇
2
+ k
2 )ψ 1 = 0
( 2 )
Further writing
k
2
1 = k
2
+ k
2
0
where k
2
0 =
2μV
2
and V = −V 0
The solutions are found to be
ψ 2 = A sin k 1 r
(3)
ψ 1 = B sin(kr + δ 0 )
( 4 )
ψ 1 (r ) is the asymptotic solution at large distances with the boundary condition
ψ 1 (0) = 0
Matching the solutions (3) and (4) at r = R both in amplitude and first
derivative,
A sin k 1 R = B sin(k R + δ 0 )
( 5 )
Ak 1 cos k 1 R = Bk cos(k R + δ 0 )
( 6 )
Dividing one equation by the other, and setting k 1 cot k 1 R =
1
D
, and with
simple algebraic manipulations we get
tan δ 0 = (k D − tan k R)(1 + k D tan k R)
−1
(7)
The phase shift δ 0 determined from (7) is a multivalued function but we are
only interested in the principle value lying within the interval −
π
2
≤ δ 0 ≤
π
2
.
For small values of the energy of the relative motion