Download Quantum wave mechanics

UNIVERSITÀ
DEGLI STUDI
Quantum scattering calculated easily
DI TRENTO
G.P. Karwasz 1,2 H. Nowakowska3
1Dipartimento
di Fisica, Università di Trento, 38050 Povo, Italy and Pomeranian Pedagogical Academy, 76-200 Slupsk, Poland
now at: Institut für Chemie-Physikalische und Theoretische Chemie, Freie Universität Berlin, 14195 Berlin
3 The Szewalski Institute of Fluid-Flow Machinery Polish Academy of Science, 80-952 Gdansk, Poland
Electrons and waves
1. Wave phenomena are characterized
by interference. The impression of an
acute dissonance happens when beats
are below 50 Hz in frequency.
2. Quantum mechanics is another example of wave interference.
In a scattering processes, the monochromatic, well-collimated
beam of particles corresponds to a plane de Broglie wave
Ψ0 = exp(ikz), with k being the wave number.
3. Following Huyghens’ principle,
the scattering center acts as
a source of spherical wave Ψ’ = exp(ikr).
4. Obviously, the scattered wave need
not be perfectly spherical, so we add an
angular factor Ψ’ = f (θ) exp(ikr).
Angular distribution of scattered electrons are complicated functions of energy.
Ramsauer’s effect
Carl Ramsauer, in Gdansk, 1921 was the first
who showed that electrons behave like waves:
at some energies the gases like Ar, Kr become
transparent to them: the cross section shows a
minimum. This is a wave-like effect.
In 1931, in Berlin- Reinickendorf, C. Ramsauer
and R. Kollath measured angular distribution
of scattered electrons, confirming
Quantum wave mechanics.
Coleman (1980)
Sinapius (1980)
Kauppila (1976)
Canter (1974)
Charlton (1984)
Stein (1992)
Present
2
Total cross section (10 m )
Positron scattering
FIRST ACCELERATOR
-20
Positrons are electrons with a positive
charge. A long discussion lasted, if
positrons also show Ramsauer minimum.
New measurements from Trento showed
surpsisingly that the (integral) cross
section reamains constant, like predicted
by Classical Mechanics for hard sphere
scattering.
REMODERATOR STAGE
10
9
8
7
Nakanishi (1986)
Gianturco (1993)
McEachran (1979)
6
5
4
3
Argon (Ar)
INJECTION OPTICS
2
1
DEFLECTOR
10
Positron energy (eV)
The scattering amplitude is given by:



where  is the angular momentum in the collision,
=0,1,2…,
 are the ”phase shifts” of the respective partial waves
and P(cos ) are Legendre polynomials
The integral cross-section is given by
4 
 int  2  2  1sin 2 
k  0
Quantum or classical?
A “small” problem arises: if scattering is
classical, angular distributions should be
uniform in angle. This is not the case!
To calculate angular distributions, the
plane wave is developed into spherical,
partial waves and a change in the phase
of the scattered wave due to the
potential is obtained from Schrödinger’s
equation. But the scattering potential is
unknown…
0.5
2
 0
-20
 2  1exp 2i   1P cos 
Differential cross section (10 m )
1
f   
2ik
5eV
8.7 eV
15 eV
0.4
0.3
0.2
0.1
0.0
0
A module in EXCELL
has been constructed allowing
to adjust phase shifts.
Now, we can reproduce any
angular distribution with a few
partial waves!
20
40
60
80
100
120
140
160
180
Scattering angle (deg)
The model shows that angular distributions
can be reproduced, and integral ones also,
but phase shifts are somewhat artificial.
An inelastic process must be added!
Why Quantum Mechanics adjusts itself to a classical result?