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Quantum Mechanics-II (PH-519)
M. Sc Physics, 3rd Semester
Contents of Course:
Relativistic Quantum Mechanics
Theory of ScatteringLecture 1
Quantum Mechanics, concept and applications by
Introduction to Quantum Mechanics by D.J.
, Quantum Mechanics
Introductory Quantum Mechanics,
Scattering involve the interaction between
incident particles (known as projectile) and target
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 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)
Particles incident into area d
σ 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 aremeasured 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
Elastic Scattering : KE remain conservede.g. (1) Rutherford scattering experiment: reveal substructure of Atom.(2) Electron proton 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
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(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
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
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
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 .
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 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)
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 particleand 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. 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.
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) respectivelyand will get corresponding current densities.
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 thedifferential cross-section reduces to the finding of scatteringamplitude.