Scattering1

Quantum Mechanics-II (PH-519)

M. Sc Physics, 3

Semester

Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: [email protected]

Contents of Course:



Scattering Theory



Perturbation Theory

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}

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 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, dN, scattered per unit

time into the solid angle dΩ is proportional

(i) Incident flux J

: It is equal to number of

incident particles per unit area per unit time.

(ii) Solid angle

dN = J

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 .

•

Scattering experiments are performed in lab frame

but calculations are easier in centre of mass frame

•

Total cross-section is independent of frame of

Elastic Scattering : KE remain conserved

e.g. (1) Rutherford scattering experiment: reveal

substructure of Atom.

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.

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

(iv) Interactions between the particles is described

by the P.E. V(r

– r

) 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).

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

Scattering in Quantum Mechanics:

We consider the scattering between two spin-less

and non-relativistic particles of masses m

and m

.

During scattering particles interact and if the

interaction is time independent then we write

the following wave function for the system,

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

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

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

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.

After scattering the total wave function is superposition of incident wave function and scattered wave function,

---(9)

We shall now show that

For this first we write flux densities corresponding to Incident and scattered wave. These are

---(10)

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

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

To find the scattering amplitude we shall use two

techniques.

(1)Born Approximation