Transcript - NIT Jalandhar

M. Sc Physics, 3rd Semester
Quantum Mechanics-II (PH-519)
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail: [email protected]
https://sites.google.com/site/karvindk2013/
Contents of Course:
Scattering Theory
Perturbation Theory
Relativistic Quantum Mechanics
Theory of Scattering
Lecture 1
Books Recommended:
Quantum Mechanics, concept and applications by
Nouredine Zetili
Introduction to Quantum Mechanics by D.J. Griffiths
 Cohen Tanudouji, Quantum Mechanics II
 Introductory Quantum Mechanics, Rechard L. Liboff
Scattering: Scattering involve the interaction between
incident particles (known as projectile) and target
material.
Play an important role in our understanding of the
structure of particles.
Reveal the substructures e.g. atom is made of
nucleus with electrons revolving around it.
The nucleus consists of proton and neutron which
are further composed of quarks.
The picture of scattering is as follows: We have a
beam of particles incident on the target material.
After collision or interaction of incident particles
with the target material, they get scattered.
The number of particles coming out varies from
one direction to other.
The number of particles, dN, scattered per unit
time into the solid angle dΩ is proportional
(i) Incident flux Jinc : It is equal to number of
incident particles per unit area per unit time.
(ii) Solid angle
dN =
Jinc dΩ
Differential
cross-section
------ (1a)
or
-------(1b)
•Particles incident into area dσ scatter into solid angle dΩ
The total cross section (σ)can be written by integrating
Eq. (1) over all solid angles i.e.
------(2)
In above Eq. we used
.
Differential cross-section has the units of area and are
measured in barn
• Scattering experiments are performed in lab frame
but calculations are easier in centre of mass frame
• Total cross-section is independent of frame of
Reference but differential cross-section depend upon
frames of reference since scattering angle is frame
dependent.
Elastic Scattering : KE remain conserved
e.g. (1) Rutherford scattering experiment: reveal
substructure of Atom.
(2) Electron proton scattering
Inelastic scattering: KE does not remain conserved
but total remain conserved
At high energy of incident beams, the KE energy
may be converted into other particles.
e.g. Deep inelastic scattering
We shall consider Elastic Scattering and assume
(i) No spin of particles
(ii) we consider pointless particles i.e. no internal
structure and hence no KE energy will be
transferred to internal constituents
(iii) Target is thin enough so no multiple scattering
(iv) Interactions between the particles is described
by the P.E. V(r1 – r2) which is depend upon relative
position of particles only.
This help to reduce problem to centre of mass system
in which two body scattering problem will reduce to
study to the scattering of reduced mass μ by the
potential V(r).
e.g. Nucleon-nucleon scattering can be studied
under above assumptions
Recall that while discussing the solutions of
Schrodinger’s equation for bound states, the
wave function vanishes at large distances from
the origin and energy levels form discrete
set.
However, here in case of scattering, we shall
study the solutions of Schrodinger equation in
which energy is distributed continuously and
wave function will not vanish
at large distances.
Scattering in Quantum Mechanics:
We consider the scattering between two spin-less
and non-relativistic particles of masses m1 and m2.
During scattering particles interact and if the
interaction is time independent then we write
the following wave function for the system,
-----(3)
where ET is total energy.
is solution of time independent Schrodinger
Eq.
---------(4)
is potential representing interaction between two
particles.
Note that if the interaction between two particles is function of
relative distance between them only then Eq. (4) can be
reduced to two decoupled equations. One is for centre of mass
(M = m1+m2) and other is for reduced mass
which moves in potential V .
Corresponding to reduced mass which moves in potential
V(r), we have following Schrödinger Eq.
------------(5)
Our scattering problem is reduced to the problem of finding
solution of above Eq (5). Eq. (5) describe the scattering of
particle of mass μ from a scattering center represented by
potential V(r). Suppose V(r) has a finite range say a.
Within range a particle interact with the potential of target,
However beyond range a, V(r) = 0. In this case Eq. (5)
become
-----------(6)
Beyond range a , the particle of mass μ behave as free
particle and can be described by plane wave
-----------(7)
where is wave vector associated with incident particle
and A is normalization factor. Before interaction with
target particle, the incident particle behave as
free particle with momentum
When the incident wave, described by Eq. (7), interact with
target, we have the scattered wave or outgoing wave. The
scattered wave amplitude depend upon direction in which it is
detected. The scattered wave is written as
--------(8)
(Note that for isotropic scattering, the scattered wave is
Spherically symmetric having form
).
In Eq. (8),
is scattering amplitude. It gives you the
probability of scattering in a given direction.
is wave vector associated with scattered wave.
After scattering the total wave function is superposition of
incident wave function and scattered wave function,
--------(9)
Note that angle between
and
or
and is zero.
joins the particle of mass μ and scattering center V(r).
We shall now show that
For this first we write flux densities corresponding to
Incident and scattered wave. These are
-----(10)
--------(11)
We use Eq. (7) and (8) in Eq. (10) and (11) respectively
and will get corresponding current densities.
We get,
-------(12)
The number of scattered particles
into solid angle
in direction
and passing through area
is written as
-----------(13)
Using (12) in (13), we get
------(14)
Using Eq. (14) and also definition of Jinc from (12), in Eq.
we get
----(15)
where normalization constant is taken as unity. Also for
elastic scattering k0 = k. Thus we have
------------(16)
From above Eq. We observe that the problem of finding the
differential cross-section reduces to the finding of scattering
amplitude.
To find the scattering amplitude we shall use two
techniques.
(1)Born Approximation
(2) Partial wave analysis