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Quantum Mechanical Cross Sections
In a practical scattering experiment the observables we have on hand are
momenta, spins, masses, etc.. We do not directly measure the distances
between the projectile and the target nucleus.
pi
pf
We start with an initial system of a target in
state |A> and the projectile in momentum state
|pi>. After the reaction the residual product is in
state |B> and the outgoing fragment in
momentum state | pf>.
We call the initial state |a>=|Api> and the final state |b>=| Bpf>. We know from
experience that interactions cause changes in momenta, and in general, all other
changes a system may experience. In classical physics we call the Hamiltonian the
generator of time evolution. This is a statement of dynamics. The same concept
applies in quantum mechanics. The Hamiltonian, H, governs the time evolution of a
quantum system. Equation (1) is the Schroedinger equation in ket notation.
 | 
H |   i
t
(1)
1
We define the time evolution operator U(t,t0) such that a system evolves
from some initial configuration, |a> to a final configuration |c> according to
|c; t> = U(t,t0) |a; t0>. U(t,t0) is a function of the Hamiltonian. The
relationship between U(t,t0) and the Hamiltonian is worked out in books on
quantum mechanics (ref. 1,2). We will assume that the target and projectile
have some internal Hamiltonians HA and Hp and that the target and
projectile interact via an interaction V. The total Hamiltonian is
H = HA + Hp + V = H0 + V. Where H0 = HA + Hp , is called the free Hamiltonian.
We assume that V becomes negligibly small for large separations between
the projectile and target. The total time evolution operator can be written as
( ref.2)
U(t,t0) = U0(t,t0) UI(t,t0) . In this expression U0(t,t0) is the time evolution operator
derived from the free Hamiltonian and UI(t,t0) depends on both H0 and V. It
is expressed in the integral equation
t
i
U I (t , t0 )  1   dt ' H I (t ' , t0 )U I (t ' , t0 ),
 t0
(2)
H I (t ' , t0 )  U 01 (t ' , t0 )V (t ' , t0 )U 0 (t ' , t0 ),
(3)
H 0 (t 't0 )
U 0 (t ' , t0 )  exp( i
),

(4)
2
We assume that the free Hamiltonian is time independent. Then U0(t’,t0) in
equation (4) is the time evolution operator due to H0. If the states |a> and |b>
are eigenstates of H0, then the time evolved states of |a> and |b> are
Ea
(t  t0 )) | a; t0 ,

E
| b; t  U 0 (t , t0 ) | b; t0  exp( i b (t  t0 )) | b; t0 ,

| a; t  U 0 (t , t0 ) | a; t0  exp( i
(5a)
(5b)
The time evolved state |c; t> = U(t,t0) |a; t0> is given by ( ref. 2),
| c; T / 2  U (T / 2,T / 2) | a;T / 2 
  Sba (T / 2,T / 2)U 0 (T / 2,T / 2) | b;T / 2 , (6)
b
Equation (6) tells us that at time t=-T/2 the system is initially in state |a>. At time T/2
the system has evolved into state |c;T/2> which consists of a sum of possible
eigenstates |b;-T/2>, each weighted with an amplitude given by Sba , which is time
dependent. If we let T/2 and –T/2 go to infinity, then Sba is called the “S – matrix”. The
S-matrix must be properly normalized (See ref. 2 chapt. 3). Particle decay rates and
cross sections can be calculated once the S-matrix is known.
3
From our discussion in lecture 2 we determined that the scattering rate is
Nba  Nt t
t  T / 2  (T / 2)  T
L


j   pv
L
L
Nba | Sba (T / 2,T / 2 | Nt N p
We want to connect this
derivation to a quantum
calculation. The probability that
the initial state a will make the
transition to b is | Sba|2 . If N t is
the number of targets in the
box of sides L, and Np the
number of projectiles which
have traversed the box then
the number of transitions from
state a to b is :
2
 ˆ N p
   p v  S  3 v, Ŝ is the surface normal
L
We can solve for  ,
| Sba (T / 2,T / 2) |2 N p
 
N p
T
v
3
L
4
| Sba (T / 2,T / 2) |2 L3
  lim T 
, (7)
T
v
In an actual calculation the S-matrix will contain terms that cancel the L3 term in (7).
In a non-relativistic calculation v = p/m, whereas for the relativistic case v = p/E.
Examples of the differential cross section  are:
Elastic scattering of a particle into a solid angle W.
d
 
W
dW
Projectile scattering into a solid angle W, and momentum bite p.
d 2
 
Wp
dWdp
Cross section for 3 body final state, as in 3He(e,e’p)2H, ( 2 bound nuclear states)
d 5
 
We W p pe
dW e dW p dpe
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Multi-body final state where the electron and proton are detected, in 3He(e,e’p)X ,
d 6
 
We W p pe p p
dW e dW p dpe dp p
Examples of calculating the S-matrix ( ref. 2 chapter 3). We want to calculate
| Sba (T / 2,T / 2) |2
lim Wba (T ) 
T 
T
since
U (t , t0 )  U 0 (t , t0 )U I (t , t0 ), so, | c, t  U 0 (t , t0 )U I (t , t0 ) | a, t0 
From eqn(6)
 b' , t | c, t  b' , t | U 0 (t , t0 )U I (t , t0 ) | a, t0 
and  b' , t | b' , t 0 | U 0* (t , t0 ), so since U 0* (t , t0 )U 0 (t , t0 )  1
 b' , t | c, t  b' , t 0 | U I (t , t0 ) | a, t0   Sba (t , t0 )  b' , t 0 | b, t0 
b
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The states are assumed to be orthonormal so we get
Sba (T / 2,T / 2)  b | U I (T / 2,T / 2) | a , (8)
where we abbreviate | a   | a, - T/2 
Substituting eqn (2) into eqn (8) yields
i T /2
Sba (T / 2,T / 2)  b | {1   dt ' H I (t ' ,T / 2)U I (t ' ,T / 2)} | a , (9)
 T / 2
Equation (9) implicitly includes the time evolution operator, so solutions to (9) are
obtained through successive iterations, for example,
i t'
U I (t ' ,T / 2)  1  
dt" H I (t" ,T / 2)U I (t" ,T / 2)
 T / 2
If we substitute this expression into eqn (9) we obtain
i T /2
Sba (T / 2,T / 2)  b | {1   dt ' H I (t ' ,T / 2)}{
 T / 2
i t'
1   dt" H I (t" ,T / 2)U I (t" ,T / 2)} | a 
 T / 2
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(1)
Sba (T / 2,T / 2)  b | a   Sba
(T / 2,T / 2) 
t'
i 2 T /2
 b | (- )  dt ' H I (t ' ,T / 2)  dt" H I (t" ,T / 2)U I (t" ,T / 2) | a 
T / 2
 T / 2
i T /2
where S (T / 2,T / 2)    dt '  b | H I (t ' ,T / 2) | a , (10)
 T / 2
(1)
ba
Eqn (10) is the first approximation, or Born approximation, in the perturbation
series expansion of the S-matrix. The expansion can be carried out to higher
orders as can be seen. The second order term in the expansion is
t'
i 2 T /2
S (T / 2,T / 2)  (- )  dt '  dt"  b | H I (t ' ,T / 2) H I (t" ,T / 2) | a 
T / 2
 T / 2
( 2)
ba
We will focus on the first term, equation (10), as an approximation for the Smatrix.
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From equation (3) we obtain HI.
 b | H I (t ' , t0 ) | a  b | U 01 (t ' , t0 )V (t ' , t0 )U 0 (t ' , t0 ) | a 
and
i
U 0 (t ' , t0 ) | a  exp(  Ea (t 't0 )) | a 

and
i
1
 b | U 0 (t ' , t0 )  b | exp( Eb (t 't0 ))

i T /2
i
S (T / 2,T / 2)    dt ' exp( ( Eb  Ea )(t 'T / 2))  b | V (t ' ,T / 2) | a 
 T / 2

(1)
ba
Consider the case where the interaction V is time independent. Then the time
integral only includes the exponential term. As we let T go to infinity the time
integral becomes a delta function in energy. However, we will take the limit of T to
infinity a bit later in the calculation.
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T /2
i
I (T / 2,T / 2)   exp( ibaT / 2)  dt exp( ibat )

T / 2
where  ba  ( Eb  Ea ) / 
sin(  baT / 2)
i
I (T / 2,T / 2)   exp( i baT / 2)T
, (11)

 baT / 2
where we multiplied the top and bottom of (11) by T.
sin(  baT / 2)
i
S (T / 2,T / 2)   exp( i baT / 2)T
 b | V | a , (12)

 baT / 2
(1)
ba
Substituting (12) into (7) we get for the differential cross section,
T 2
4( )
2
sin
( baT / 2)
L3
2
2
  2 | Vba | lim
2
T 
T
v
(

T
/
2
)
ba
2( )
2
10
We now take advantage of eqn 5.6.31 in ref. 1,
sin 2 (x)
lim
  ( x), where   T/2
2
 
x
And we obtain for the cross section
 (1)
L3
 2 | Vba |2 2 (ba ), (13)
v
We see that conservation of energy automatically appears if the time interval T
is long enough. We will use (13) to calculate the quantum mechanical cross
section in the first approximation (Born approximation) of the S-matrix. Note
that the matrix element is,
Vba  b | V | a 
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Cross section for potential scattering
In a practical scattering situation we have a finite acceptance for a detector
with a solid angle W. There is a range of momenta which are allowed by
kinematics which can contribute to the cross section. The cross section for
scattering into W is then obtained as an integral over all the allowed
momenta for that solid angle. In the case of potential scattering we will
assume only elastic scattering is allowed. This means that there is a change
in momentum between the incoming particle and outgoing particle. We are
using periodic boundary conditions, or box normalization, ( see ref. 2) for
which the incoming and outgoing states are plane wave states. The
incoming wave |a> is
 


1
a (r )  r | a  3 / 2 exp( ik a  r ), (14a)
L
2nax
and k ax 
, nax  0,1,2,..., (15a)
L
and the same for the y and z components .
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The outgoing state |b> is
 


1
b (r )  r | b  3 / 2 exp( ikb  r ), (14b)
L
2nbx
and k bx 
, nbx  0,1,2,..., (15b)
L
and the same for the y and z components .
All possible outgoing states |b> which can be accepted by the solid angle
must be included in the summation.
3
L
  2
v
| b | V | a |
nx n y nz W
2
2 (ba )
We next substitute the matrix element into the above equation. This will be an
integral over all relevant degrees of freedom needed to describe the states a
and b. For a position dependent interaction and the plane waves above, the
integration is over configuration space.
13




 b | V | a   d r ' d r  b | r  r | V | r '  r ' | a 
3
3
But we will consider only potentials diagonal ( local) in configuration space.
 
  
V (r , r ' )  V (r ) (r  r ' ), so

 


1
1
V(q )   dr 3 3 exp( iq  r )V (r )  3 f (q ), where
L
L





q  kb  k a , is the " momentum transfer" , (p  k)
Notice that the matrix element is proportional to the Fourier transform of the potential.

kb
q

ka

q
L3
  2
v
 2
1
| f (q ) | 2 (ba ), (16)

6
nx n y nz W L
14
1
  2
v
 2
1 1 1
( )( )( ) | f (q ) | 2 (ba ), (17)

n x n y n z W L L L
In eqn (17) we note the sum over possible n values. The smallest change
possible in n is dn = 1. We want to convert the sum in (17) into an integral over
wave numbers.
And from eqns (15) we realize that
dk x dk y dk z dnx dn y dnz 1 1 1


2 2 2
L L L
LLL
dk x dk y dk z
1 1 1
( )( )( )  
, (18)

2 2 2
n x n y n z W L L L
k x k y k z W
dk x dk y dk z k 2 dkdWk
dk 3


3
3
3
(2 )
(2 )
(2 )
15
We substitute (18) into (17) an obtain
1
  2
v
 dW
Wk

k
k x k y k z Wk
 2
dkk 2
| f (q ) | 2 ( ba ), (19)
3
(2 )
1
and  ( ba )   ( ( E (kb )  E (k a ))), (20)

Note that in the delta function in (20) we have functions ( the energies) of the
momenta k, whereas the variable of integration is k. We use the following
property of functions of the delta function.
 ( f ( x))  
xi
 ( x  xi )
df
|
| xi
dx
, where x i are the zeroes of f(x)
In the case of potential scattering, or elastic scattering, the zeroes occur for
k = kb = ka . The derivative of the energy with respect to the wave number k is
16
d E (k ) 1 d
k p
2
2

(k )  m 
 v
dk 
 dk
E E
Integrating over k, the delta function selects the value k = ka which we will
simply call k, the magnitude of the initial wave number. Thus,
 2
2 W k k 2
  2 2
| f (q ) |
3
v  (2 )
 2
d

1
k2

 2 2
| f (q ) | , (21)
2
dW k Wk  v (2 )
We are now in a position to evaluate the differential elastic cross section using
eqn (21) for any potential V. We will need to find the Fourier transform of the
potential as a function of q.
17
(1) “Modern Quantum Mechanics – Revised Edition”, J. J. Sakurai,
Addison-Wesley, 1994
(2) “Relativistic Quantum Mechanics and Field Theory”, Franz Gross,
John Wiley & Sons, 1993
18