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Quantum Mechanics-II (PH-519)

M. Sc Physics, 3rd Semester

Dr.

Arvind

Kumar

Physics Department

NIT

Jalandhar

e.mail

:

iitd.arvind@gmail.com

https://sites.google.com/site/karvindk2013/

Slide2Contents of Course:

Scattering Theory

Perturbation Theory

Relativistic Quantum Mechanics

Slide3Theory of ScatteringLecture 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

Slide4Scattering:

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.

Slide5The 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.

Slide6Slide7

The number of particles, dN, scattered per unit time into the solid angle dΩ is proportionalIncident flux Jinc : It is equal to number of incident particles per unit area per unit time.(ii) Solid angle dN = Jinc dΩ ------ (1a)or -------(1b)

Differential

cross-section

Slide8Particles incident into area d

σ scatter into solid angle dΩ

Slide9The 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 aremeasured in barn

Slide10Scattering 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.

Slide11Elastic Scattering : KE remain conservede.g. (1) Rutherford scattering experiment: reveal substructure of Atom.(2) Electron proton scattering

Slide12Inelastic scattering:

KE does not remain conservedbut total remain conservedAt high energy of incident beams, the KE energy may be converted into other particles.e.g. Deep inelastic scattering

Slide13We shall consider Elastic Scattering and assume

No spin of particles

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

Slide14(iv) Interactions between the particles is described

by the P.E. V(r

1

– r

2

) 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

Slide15Recall 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.

Slide16Scattering 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.

Slide17is solution of time independent Schrodinger

Eq. ---------(4)

is potential representing interaction between two

particles.Note that if the interaction between two particles is function ofrelative distance between them only then Eq. (4) can bereduced to two decoupled equations. One is for centre of mass(M = m1+m2) and other is for reduced mass which moves in potential V .

Slide18Corresponding to reduced mass which moves in potential V(r), we have following Schrödinger Eq. ------------(5)Our scattering problem is reduced to the problem of findingsolution of above Eq (5). Eq. (5) describe the scattering ofparticle of mass μ from a scattering center represented bypotential 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)

Slide19Beyond 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 particleand A is normalization factor. Before interaction with target particle, the incident particle behave as free particle with momentum

Slide20Slide21

When the incident wave, described by Eq. (7), interact with

target, we have the scattered wave or outgoing wave. Thescattered 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 theprobability of scattering in a given direction. is wave vector associated with scattered wave.

Slide22After 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).

Slide23We 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) respectivelyand will get corresponding current densities.

Slide24We 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)

Slide25Using 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 thedifferential cross-section reduces to the finding of scatteringamplitude.

Slide26Slide27

To find the scattering amplitude we shall use two

techniques.

Born Approximation

(2) Partial wave analysis

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