M. Sc Physics, 3rd Semester Quantum Mechanics-II (PH-519) Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com 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 Incident 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

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 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

(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. Born Approximation (2) Partial wave analysis