Quantum Mechanics Lecture on Scattering Theory and Cross Section Analysis
Explore the concepts of scattering theory, phase shifts, optical theorem, and Born approximation in quantum mechanics. Understand the differential scattering cross-section and probability of particle scattering through spherically symmetric interaction potentials.
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PHY 741 Quantum Mechanics 12-12:50 AM MWF Olin 103 Plan for Lecture 31: Chap. 19 in Shankar: Scattering theory 1. Scattering cross section in terms of phase shifts 2. Optical theorem 3. Born approximation 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 1
11/15/2017 PHY 741 Fall 2017 -- Lecture 31 2
11/15/2017 PHY 741 Fall 2017 -- Lecture 31 3
Scattering geometry ikr e ( ) k f k r ie r Differential scattering cross section Probability of particle scattering Incident flux of particles d d = 2 ( , ) k r = f 2 2 4 k sin( ( )) l ( ) E i * r = ( ) ( ) e E Y Y l lm lm k lm 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 4
Some details: * ( ) k = k r i l r It can be shown that: 4 ( ) ( ) e i j kr Y Y l lm l m lm In presence of spherically symmetric interaction potential ( ): ( ) ( ( ) ) E El lm lm Differential equation for rad ial function + + + Outside the range of ( ); when ( ( ) ( ) El l l l l R r A j k r B y kr = + = V r = r r R r Y 2 2 2 r dr ( 1) d d l l = ( ) ( ) 0 V r E R ) j kr r El 2 2 2 m dr r 0: ( V r V r ( ) ) sin N ( ) cos ( ) y k r l l l l l 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 5
Denote by a radius outside the range of ( ). We can calculate the log-deri ( ) ln ( ) ( ) El dr R r D V r vative: dR r El dr ( ) d R r = El ( ) E L l r D = It follows that: ( ) ( ( ) E ) ) ( ' ' ) ) L L E j kD y kD kj kD ky kD ( ) = tan ( ) E l l l l ( ( l l l 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 6
From the assymptotic form of the = Bessel fun ctions: 4 k r k sin( ( )) l ( ) E i * r ( , ) f ( ) ( ) e E Y Y l lm lm k lm d d 2 k r = ( , ) f Total scattering cross section: d E d d 4 k ( ) = = + n ( ( )) l 2 ( ) 2 1 si l E 2 l 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 7
Interesting identities; "optical theorem" + ( ) 2 1 l l ( ) r k k r = * Note that: ( ) Y Y P lm lm l 4 = m l k r 4 k = where ( ) is a Legendre polynomial where P (1) 1 P l l k r k = ( ) E i * r ( , ) f sin( ( )) l ( ) ( ) e E Y Y l lm lm lm 4 k ( ) = + 2 ( ) E 2 1 sin ( ( )) l E l 2 l Note that: Imaginary part of forward scattering is proportional to the total scattering cross section. ( ) ( ) k k = = r ( ) E f 4 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 8
The phase shift analysis of scattering theory provides a convenient mechanism to relate experiment to the interaction potential Example -- 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 9
This minimum in is not seen in e-He and e-Ne scattering. e V r r 2 It can be explained by ( ) due to atom polarization. 4 2 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 10
Approximate treatment of scattering Born approximation In this treatment, we use the notions of perturbation th + = eo y r 0 1 H H H 2 = 0 2 H 2 m = 1 ( ) r H V 0 In this case, the relevant eigenstates of are plane waves. H 2 mE = = = k r 0 0 E 0 E 0 E i where H E e k 2 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 11
Equation for first order wavefunction: = r ( Note that for ) 0 1 0 ( ) ( ) H E V r 2 ( ) = 2 ( , ', ) r r r r ' E G E 2 m r-r' ik 2 m e = ( , ', ) r r G E 2 r-r' 4 = G 1 3 0 ( , ', ) ( ') E V r r r ( ) r d r r-r' i k 2 m e r-r' k r k r 3 ' i i ( ') r e d r V e 2 4 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 12
r -r' ik 2 m e r-r' k r k r 3 ' i i ( ') e d r V r e 2 4 ' r r r-r' For ', r r r ikr 2 m e r e r r k r k r 3 ' V r ' i ik i ( ' ) e d e 2 4 r ik r 2 m e ( ) e r r k ' V r i k k r 3 i ( ') e d r 2 4 r Scattering amplitude in the Born appr 2 ( 4 oximat n: io m ( ) k r r , ) k r r e k ' V r i = 3 ( ') f d 2 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 13
Example screened Coulomb interaction kr 2 Ze r = r ( ) V r e k 2 m ( ) , ) k r r e r r k ' V r i k = 3 ( ( ') f d 2 4 2 2 m Ze K r = ' sin( ') d r e Kr 2 0 2 2 m Ze = ( ) 2 + 2 2 K = = r k where 2 sin( k / 2) K k 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 14
Example screened Coulomb interaction -- continued kr k 2 s k = 2 Ze r = r ( ) V r e = r k in ( / 2 ) K k 2 2 m Ze , )= k r ( f ( ) 2 + 2 2 K Differential cro ss section : 2 2 2 2 1 + d d mZe 2 k r = = ( , ) f ( ) 2 2 2 K 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 15
Example spherical well kr for 0 for V r a a 0 = k ( ) V r r 2 m ( ) , ) k r r e r r k ' V r i k = 3 ( ( ') f d 2 4 2 4 m a 4 V m = ' 'sin( r r ') 0 d Kr 2 K 0 2 V K k ( ) = sin( ) cos( ) 0 Ka Ka Ka 2 3 = = r where 2 sin k ( / 2) K k 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 16
Example spherical well continued kr k 2 s k = for 0 for 2 m V r a a = r k in ( / 2 ) K k 0 = ( ) V r r V K ( ) , )= k r ( sin( ) cos( ) f Ka Ka Ka 0 2 3 Differential cross section: 2 2 1 mV d d 2 k r 2 = = , ( ) sin( ) cos( ) f 0 Ka Ka Ka 2 6 K 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 17
Beyond the Born approximation Equation for full wavefunction: r ( ) = 0 ( ) ( ) H E V r r r r-r' ik 2 2 m e ( ) = = 2 ( , ', ) r r r r ' ( , ', ) G r r E G E E 2 r-r' 2 4 m = G 3 ( ) r ( , ', ) ( ') E V r r r d r G + 0 3 0 ( , ', ) ( ') E V r r r r ( ) d r 3 0 3 0 r ( '', ', ) ( ') E V r r r ( ) r + '' ( , '' , ) ( '' V E ) ( ' ') d r G r d r G +... 11/15/2017 PHY 741 Fall 2017 -- Lecture 31 18
