Chapter 18 Scattering in Quantum Mechanics Application of 3D Green's function Masatsugu Sei Suzuki Department of Physics SUNY at Binghamton (Date: November 16, 2010)
Max Born (11 December 1882 – 5 January 1970) was a German born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s. Born won the 1954 Nobel Prize in Physics (shared with Walther Bothe).
http://en.wikipedia.org/wiki/Max_Born ______
Julian Seymour Schwinger (February 12, 1918 – July 16, 1994) was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order. Schwinger is recognized as one of the greatest physicists of the twentieth century, responsible for much of modern quantum field theory, including a variational approach, and the equations of motion for quantum fields. He developed the first electroweak model, and the first example of confinement in 1+1 dimensions. He is responsible for the theory of multiple neutrinos, Schwinger terms, and the theory of the spin 3/2 field.
http://en.wikipedia.org/wiki/Julian_Schwinger
______18.1 Green's function in scattering theory We return to the original Schrödinger equation.
2 2 (r) V (r) (r) E (r) , 2m k or
2 2m 2m ( 2 Ek ) (r) 2 V (r) (r) .
We assume that
2 E k 2 , k 2m
2m f (r) 2 V (r) (r) ,
2 2 Lr k .
2 2 Lr (r) ( k ) (r) f (r)
Suppose that there exists a Green's function G(r) such that
2 2 (r k )G(r,r') (r r') , with
exp(ik r r') G(r,r') . 4 r r'
Then (r) is formally given by
2m exp(ik r r') (r) (r) dr'G(r,r') f (r') (r) dr' V (r') (r') , 2 4 r r' where (r) is a solution of the homogeneous equation satisfying
(2 k 2 )(r) 0 ,
1 (r) r k exp(ik r) , (plane wave) (2 )3 / 2 with
k k
18.2 Born approximation
M
r r - r
uˆ r 0P
L
r |r - r
r' rˆ
r
Here we consider the case of () (r)
r r' r r'er
k' ker
ik rr' e eik(rr'er ) eikreik'r' for large r.
1 1 r r' r
Then we have
1 2m 1 eikr () (r) exp(ik r) dr'eik'r'V (r') () (r') 3/ 2 2 (2 ) 4 r or
1 eikr () (r) [eikr f (k',k)] (2 )3 / 2 r
The first term: original plane wave in propagation direction k. The second term: outgoing spherical wave with amplitude, f (k',k) ,
1 2m f (k',k) (2 )3 / 2 dr'eik'r'V (r') () (r') . 2 4
The first Born approximation:
1 2m f (k',k) dr'eik'r'V (r')eikr' 2 4 when () (r) is approximated by
1 () (r) eikr . (2 )3/ 2
18.3 Differential cross section
d Detector V(r)
z 0
ikz f (,) e k eikz r incident spherical plane wave wave
d We define the differential cross section as the number of particles per unit time d scattered into an element of solid angle d divided by the incident flux of particles.
The particle flux associated with a wave function
1 1 (r) r k eikr eikz k (2 )3 / 2 (2 )3 / 2 is obtained as
1 k v N J [ * (r) (r) (r) * (r)] z z 2mi k z k k z k (2 )3 / 2 m (2 )3 / 2
unit area
z k v rel relative velocity Fig. Note that = m in this figure.
k volume = 1 m
ikz 2 e 1 means that there is one particle per unit volume. Jz is the incident flux (number of particles) of the incident beam crossing a unit surface perpendicular to OZ per unit time.
The flux associated with the scattered wave function
1 eikr f ( ) r (2 )3 / 2 r is
2 1 k f ( ) J ( * * ) r 2mi r r r r r r (2 )3 / 2 m r 2
dA r 2d
2 v f ( ) v 2 N J dA r 2d f ( ) d r (2 )3 / 2 r 2 (2 )3 / 2
(the number of particles which strike the opening of the detector per unit time)
Detector
r dS d 2 dS r d
The differential cross section
d N 2 d f ( ) d d N z or
2 f ( )
First order Born amplitude:
1 2m 1 2m f (k',k) (2 )3 k'V k dr'ei(k k')rV (r') 2 2 4 4 which is the Fourier transform of the potential with respect to q, where
q k k' : scattering wave vector.
For a spherically symmetric potential, f (k',k) is a function of q.
q 2k sin( ) , 2 where is an angle between k’ and k (Ewald’s sphere). Fro simplicity we assume that ’ is an angle between q and r’. We can perform the angular integration over ’.
1 2m f (1) ( ) dr'eiqrV (r') 4 2 1 2m dr' d 'eiqr'cos ' 2r'2 sin 'V (r') 2 4 00
Note that
2 d 'eiqr'cos ' sin ' sin(qr') 0 qr'
Then
1 2m 2 f (1) ( ) dr'2r'2 V (r') sin(qr') 2 4 0 qr'
1 2m dr'r'V (r')sin(qr') 2 q 0
The differential cross section is given by
d 2 f (1) ( ) d
18.4 Yukawa potential
Hideki Yukawa (23 January 1907 – 8 September 1981) was a Japanese theoretical physicist and the first Japanese Nobel laureate.
http://en.wikipedia.org/wiki/Hideki_Yukawa ______The Yukawa potential is given by
V V (r) 0 er , r
where V0 is independent of r. 1/ corresponds to the range of the potential.
2m 1 V 2m 1 V f (1) ( ) dr'r' 0 er' sin(qr') 0 dr'er' sin(qr') 2 2 q 0 r' q 0
Note that
q dr'er' sin(qr') 2 2 0 q
(1) 2m V0 1 f () 2 2 2 q
Since
q 2 4k 2 sin 2 ( ) 2k 2 (1 cos ) 2 so, in the first Born approximation,
2 2 d (1) 2mV0 1 f ( ) 2 2 2 2 d [2k (1 cos ) ]
Note that as 0 , the Yukawa potential is reduced to the Coulomb potential, provided the ratio V0 / is fixed.
V 0 ZZ'e2
2 2 2 d (1) 2 (2m) (ZZ'e ) 1 f ( ) 4 4 4 d 16k sin ( / 2)
2k 2 Using E , k 2m
2 2 d 2 1 ZZ'e 1 f ( ) 4 d 16 Ek sin ( / 2) which is the Rutherford scattering cross section that can be obtained classically.
18.5 Validity of the first-order Born approximation If the Born approximation is to be applicable, r () should not be too different from r k inside the range of potential. The distortion of the incident wave must be small.
2m eik rr' r () r k dr' V (r') r' () 2 4 r r'
r () r k at the center of scattering potential at r = 0.
2m eikr' eikr' 1 dr' V (r') 2 3 / 2 3 / 2 4r' (2 ) (2 ) or
2m 1 eikr' dr' V (r')eikr' 1 2 4 r'
______18.6 Lipmann-Schwinger equation
The Hamiltonian H is given by
ˆ ˆ ˆ H H0 V
where H0 is the Hamiltonian of free particle. Let be the eigenket of H0with the energy eigenvalue E,
ˆ H0 E
The basic Schrödinger equation is
ˆ ˆ (H0 V ) E (1)
ˆ ˆ ˆ Both H0 and H0 V exhibit continuous energy spectra. We look for a solution to Eq.(1) such that as V 0 , , where is the solution to the free particle Schrödinger equation with the same energy eigenvalue E.
ˆ ˆ V (E H 0 )
ˆ Since (E H 0 ) = 0, this can be rewritten as
ˆ ˆ ˆ V (E H0 ) (E H0 ) which leads to
ˆ ˆ (E H0 )( ) V or
(E Hˆ )1Vˆ 0 .
The presence of is reasonable because must reduce to as Vˆ vanishes.
Lipmann-Schwinger equation:
() ˆ 1 ˆ () k (Ek H0 i ) V
2 2 by making Ek (= k / 2m) slightly complex number (>0, ≈0). This can be rewritten as
r () r k dr' r (E Hˆ i )1 r' r' Vˆ () k 0 where
1 r k eikr , (2 )3/ 2 and
ˆ H0 k' Ek ' k' , with
2 E k'2 . k ' 2m
The Green's function is defined by
2 1 G() (r,r') r (E Hˆ i )1 r' eik rr' 2m k 0 4 r r'
((Proof))
2 I r (E Hˆ i )1 r' 2m k 0
2 2 dk'dk" r k' (E k'2 i )1 k' k" k" r' 2m k 2m or
2 2 I dk'dk" r k' (E k'2 i )1 (k'k") k" r' 2m k 2m 2 2 dk' r k' (E k'2 i )1 k' r' 2m k 2m 2 dk' eik'(rr') 2m (2 )3 2 E k'2 i k 2m where
2 E k 2 . k 2m
Then we have
2 dk' eik'(rr') I 2m (2 )3 2 (k 2 k'2 ) i 2m dk' eik'(rr') K () (r r') (2 )3 k'2 (k 2 i ) 0
In summary, we get
2m r () r k dr'G() (r,r') r' Vˆ () 2 or
2m r () r k dr'G() (r,r')V (r') r' () . 2
More conveniently the Lipmann-Schwinger equation can be rewritten as
() ˆ 1 ˆ () k (Ek H0 i ) V with
2 Gˆ () (E Hˆ i )1 , 2m k 0 and
2m ˆ () ˆ ˆ 1 ˆ 2 G V (Ek H0 i ) V
When two operators Aˆ and Bˆ are not commutable, we have very useful formula as follows,
1 1 1 1 1 1 (Bˆ Aˆ) (Bˆ Aˆ) , Aˆ Bˆ Aˆ Bˆ Bˆ Aˆ where [Aˆ, Bˆ] 0 are not commutable. We assume that
ˆ ˆ ˆ ˆ A (Ek H0 i ) , B (Ek H i )
ˆ ˆ ˆ ˆ ˆ A B H H0 V
Then
ˆ 1 ˆ 1 ˆ 1 ˆ ˆ 1 (Ek H0 i ) (Ek H i ) (Ek H i ) V (Ek H0 i ) , or
ˆ 1 ˆ 1 ˆ 1 ˆ ˆ 1 (Ek H i ) (Ek H0 i ) (Ek H0 i ) V (Ek H i ) .
For simplicity, we newly define the two operators by
Gˆ (E i ) (E Hˆ i )1 0 k k 0
ˆ ˆ 1 G(Ek i ) (Ek H i )
ˆ ˆ where G0 (Ek i ) denotes an outgoing spherical wave and G0 (Ek i ) denotes an incoming spherical wave. Note that the operator Gˆ () is slightly different from Gˆ (E i ) , 0 k
2 2 Gˆ () (E Hˆ i )1 Gˆ (E i ) . 2m k 0 2m 0 k
Then we have
Gˆ (E i ) Gˆ(E i ) Gˆ(E i )VˆGˆ (E i ) 0 k k k 0 k ˆ ˆ ˆ G(Ek i )[1VG0 (Ek i )]
Gˆ(E i ) Gˆ (E i ) Gˆ (E i )VˆGˆ(E i ) k 0 k 0 k k ˆ ˆ ˆ G0 (Ek i )[1VG(Ek i )]
Then () can be rewritten as
() ˆ ˆ () k G0 (Ek i )V ˆ ˆ ˆ ˆ () k G(Ek i )[1VG0 (Ek i )]V ˆ ˆ () ˆ ˆ () k G(Ek i )V ( G0 (Ek i )]V ˆ ˆ k G(Ek i )V k ˆ ˆ [1 G(Ek i )V ] k or
1 () k Vˆ k . ˆ Ek H i
18.7 The higher order Born Approximation From the iteration, () can be expressed as
() ˆ ˆ () k G0 (Ek i )V ˆ ˆ ˆ ˆ () k G0 (Ek i )V ( k G0 (Ek i )V ) ˆ ˆ ˆ ˆ ˆ ˆ k G0 (Ek i )V k G0 (Ek i )VG0 (Ek i )V k ...
The Lippmann-Schwinger equation is given by
() k Gˆ (E i )Vˆ () k Gˆ (E i )Tˆ k 0 k 0 k , where the transition operator Tˆ is defined as
Vˆ () Tˆ k or
ˆ ˆ () ˆ ˆ ˆ ˆ T k V V k VG0 (Ek i )T k
This is supposed to hold for any k taken to be any plane-wave state.
ˆ ˆ ˆ ˆ ˆ T V VG0 (Ek i )T .
The scattering amplitude f (k',k) can now be written as
2m 1 3 ˆ () 2m 1 3 ˆ f (k',k) 2 (2 ) k'V 2 (2 ) k' T k . 4 4
Using the iteration, we have
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ T V VG0 (Ek i )T V VG0 (Ek i )V VG0 (Ek i )VG0 (Ek i )V ...
Correspondingly we can expand f (k',k) as follows:
f (k',k) f (1) (k',k) f (2) (k',k) f (3) (k',k) ...... with
(1) 2m 1 3 ˆ f (k',k) 2 (2 ) k' V k , 4
(2) 2m 1 3 ˆ ˆ ˆ f (k',k) 2 (2 ) k'VG0 (Ek i )V k , 4
(3) 2m 1 3 ˆ ˆ ˆ ˆ ˆ f (k',k) 2 (2 ) k' VG0 (Ek i )VG0 (Ek i )V k . 4
fk ' fk '
V fk ' V G0
V G0 V
G0 V
V
fk
fk fk
Fig. Feynman diagram. First order, 2nd order, and 3rd order Born approximations.
k k is the initial state of the incoming particle and k ' k' is the final state of the incoming particle. Vˆ is the interaction.
18.8 Optical Theorem The scattering amplitude and the total cross section are related by the identity
k Im[ f ( 0)] 4 tot where
f ( 0) f (k,k) : scattering in the forward direction.
d d . tot d
This formula is known as the optical theorem, and holds for collisions in general.
Ñk s m tot
Ñk Ñk Ñk O - s m m m tot
Fig. Optical theorem. The intensity of the incident wave is k / m . The intensity of the forward wave is (k / m) (4 / m) Im[ f (0)]. The waves with the total intensity
(4/ m)Im[ f (0)] (k / m) tot is scattered for all the directions, as the scattering spherical waves.
((Proof))
1 3 2m ˆ Im[ f (k,k)] (2 ) 2 Im[ k T k ] 4 1 3 2m ˆ () (2 ) 2 Im[ k V ] 4
() k Gˆ (E i )Vˆ () 0 k , or
k () Gˆ (E i )Vˆ () 0 k , or
k () () VˆGˆ (E i ) 0 k .
Then
Im[ k Vˆ () Im[( () () VˆGˆ (E i ) () 0 k .
Now we use the well-known relation
1 Gˆ (E i ) 0 k ˆ Ek H0 i
1 [P i (E Hˆ )] ˆ k 0 Ek H0
Then
1 Im[ k Vˆ () Im[( () () [Im () VˆP Vˆ () Im () Vˆi (E Hˆ )Vˆ () ] ˆ 0 E H0
The first two terms of this equation vanish because of the Hermitian oparators of Vˆ and
1 VˆP Vˆ . ˆ E H0
Therefore,
ˆ () () ˆ ˆ ˆ () ˆ ˆ ˆ Im[ k V V (Ek H 0 )V k T (E H 0 )T k or
2k'2 2 2k'2 Im[ k Vˆ () dk' k Tˆ k' k' Tˆ k (E ) dk' k' Tˆ k (E ) k 2m k 2m
2k'2 2k 2 2k'2 (E ) ( ) 2m 2m 2m
2 2 k 2 [ (k 2 k'2 )] [ (k k')(k k'))] [ (k k')] 2m 2m m or
2 2 k' m (E ) 2 (k k') 2m k
m 2 Im[ k Vˆ () dk'k'2 d' k' Tˆ k (k k') 2 k mk 2 d' k' Tˆ k 2
Thus
1 3 2m ˆ () Im[ f ( 0)] (2 ) 2 Im[ k V ] 4 1 2m mk 2 (2 )3 ( ) d' k' Tˆ k 2 2 4 k 16 4m2 2 d' k' Tˆ k ] 4 4
Since
d 2 f (k',k) d' 2 2 1 6 4m ˆ 2 (2 ) 4 k' T k , 16 4 2 16 m ˆ 2 4 k' T k
d 16 4m2 2 d' d' k' Tˆ k tot 4 d'
Then we obtain
k Im[ f ( 0)] . 4 tot ______REFERENCE
J.J. Sakurai, Modern Quantum Mechanics, Revised Edition (Addison-Wesley, Reading Massachusetts, 1994). A.G. Sitenko and P.J. Shepherd, Lectures in Scattering Theory (Pergamon Press, Oxford, 1971). E. Merzbacher, Quantum Mechanics 3rd edition (John Wiley & Sons, Inc., New York, 1998).