Block – 2 : Angular momentum spin and scattering
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M.Sc. Final (Physics)
Block – 2
Paper - I
UNIT 3 ANGULAR MOMENTUM
UNIT 4 SPIN
UNIT 5 SCATTERING THEORY
Angular momentum, Spin and Scattering
UNIT 1 ANGULAR MOMENTUM
Structure
1.0
Introduction
1.1
Objectives
1.2
Eigen values and Eigen vectors of Angular Momentum
1.2.1
Angular momentum
1.2.2
Eigen values and Eigen vectors* of Lz
1.3
Characteristic Algebraic Relations
1.3.1 Commutation relations
1.4
Spectrum of J2, Jz
1.5
Eigen Vectors of J2 and Jz
1.5.1
1.5.2
1.6
Total angular momentum
Ladder operators
Orbital Angular Momentum and Spherical Harmonics.
1.6.1
Operators in polar form
1.6.2
1.6.3
Eigen values and Eigen functions of L z
To find Lx operator
1.6.4
Eigen values and Eigen functions of L2
1.7
Angular momentum and Rotations.
1.7.1
Commutation relations for Angular Momentum Operators
1.7.2
Eigen values and Eigen functions of Lz
1.7.3
Eigen function and Eigen values for L2
1.8
Rotation Operator
1.9
Rotational Invariance and Conservation of Angular Momentum
1.10 Rotational Degeneracy
2
Angular momentum
1.11 Let Us Sum up
1.12 Check Your Progress: The Key
1.0 INTRODUCTION
Angular momentum is one of the basic features of quantum mechanics. Its conservation
is a universal phenomenon independent of the nature of the reference frame that allows
calculation of energy values of the system under different conditions. In relativistic
domain also it holds good. Rate of change of angular momentum gives torque on the
system of particles. In the following discussions we will have exhaustive treatment for
angular momentum operator, the operator in spherical polar coordinates and finding
Eigen values and Eigen functions. Further we will notice that angular momentum is
rotational analogue for the linear momentum.
1.1 OBJECTIVES
After interacting with the material presented here students will be able to
i) Relate two kinds of angular momenta with a dynamic system
ii) Write angular momentum operator in polar form
iii) Use commutation relations
iv) Understand features of ladder operators
v) Calculate Eigen values for Lz and J
vi) Relate angular momentum to the rotation of frame of reference
1.2 EIGEN VALUES AND EIGEN VECTORS OF ANGULAR MOMENTUM
1.2.1 Angular momentum
Angular momentum L for a particle of momentum p is defined through
3
Angular momentum, Spin and Scattering
L=rxp
(To understand it classically it is to be considered as moment of momentum).
In its fully expanded form
L = iLx + j Ly + kLz
= (i x + j y + k z) x (i px + j py + k pz)
= i (y pz – z py) + j(z px – x pz) + k(x py – y px)
Here we have expressed vectors r, p and L in their components along the Cartesian axes.
We have also used the orthogonal properties of vectors viz., i x i = j x j = k x k = 0 ; i x j
= k, j x k = i, k x i = j and j x i= - k etc.
Let us concentrate on the z-component of angular momentum (equating coefficients of k)
we have
Lz = (x py – y px)
We can express this relation in operator form by writing a cap (^ symbol) on each of the
letter under consideration. Thus Lx represents an Lx operator. Operator form of L is L .
The other symbols can be Lz → Lz etc., x → x etc. and px → p x = - iħ /x etc.
We find
Lz = - iħ ( x /y - y /x)
(1)
Similarly we can find the expressions for Lx and Ly . Hint: Use symmetry to write
Lx i ( y
Ly i ( z
and
z )
z
y
(2)
x )
x
z
(3)
It is found that Eigen functions for many practical systems when expressed in Cartesian
coordinates are very cumbersome and complex in comparison to those expressed in
cylindrical polar coordinates. Hence we will express a few occasionally used Eigen
functions polar coordinates and then find corresponding Eigen values for them.
1.2.2 Eigen values and Eigen vectors* of Lz
(*Eigen functions are also called Eigen vectors)
The relation
4
Angular momentum
Lz i
indicates that the corresponding Eigen function could be = (). Writing Eigen value
equation as
Lz () = E()
A trial solution could be
() = Ae-im
i
(Aeim ) EAeim
- iħA(-im) e-im = Ae-im
Or
suggesting that
E = mħ
(4)
We find in the following section that = Ae-im
1.3 CHARACTERISTIC ALGEBRAIC RELATIONS
1.3.1 Commutation relations
When two operators A and B are written as A B and B A we have an important
relation
A B- B A = [ A,B]
(5)
The operator [ A , B ] is called commutation operator.
Example 1
To find what the operator [ x p x p x x ] represents?
We must remember that an operator without an operand hardly conveys any meaning.
Hence we write [ x p x p x x ](x) to find what the operator is like. Writing these
operators in their usual form gives
[x (iħ
( x)
( x)
) – (iħ
)x] (x) = - [x (iħ
) – (iħ
)x]
x
x
x
x
= - [x (iħ
( x)
( x)
) - iħ(x) - xi(x)
]
x
x
5
Angular momentum, Spin and Scattering
= - iħ(x)
[x (iħ
Or
) x - x(iħ
)] = iħ
x
x
[(iħ
Corrolary
) – (iħ
)x] = - iħ
x
x
(6)
Check your progress-1
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) Show that [ p,,x] = iħ
Hint p i
and terms like iħ
x =0
j
k
x
y
z
y
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ii) Find [x2,px]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Example 2
To show that the operator [x, d/dx] = -1
Let us operate this operator on , we have
x
Remember
Thus
d
d
d
d
x. = x
- - x = -
dx
dx
dx
dx
d
x =1
dx
[x, d/dx] = -
qued.
6
Angular momentum
Can you show that [d/dx, x] = 1?
Some commutation rules
To solve commutation problems we need to remember following rules so that every time
we need not worry to obtain relations from first principles. Those curious enough can
easily verify those relations from definition (first principles).
1. [(A+B),C] = [A,C] + [B,C]
2. [AB,C] = [A,C]B + A[B,C]
3. [AB,CD] = A{[B,CD]} + {[A,CD]}B
4. [A,B] = - [B,A]
(7)
1.4 EIGEN VALUES (OR SPECTRUM) OF JZ AND J2
Let us assume that the simultaneous Eigen functions for J2 and Jz are (,m) such that
J2 (,m) = ħ2(,m)
and
Jz (,m) = ħm(,m)
to find values of and m, we use ladder operators with some well known relations given
below:
[J+,Jz] = - iħJ+
[J-,Jz] = ħJ[J+,J-] = - 2ħJz
and
J+ † = J- and J+ = J-†
(8)
( † is spelt as dagger and so the above quantities are spelt as J plus dagger equals J minus
etc. Dagger represents complex conjugate of the quantity over which it is super scripted.)
We will also use
J+J- = J2 – Jz2 + ħJz
and
J-J+ = J2 – Jz2 - ħJz
(9)
(Would you like to verify these relations by actually substituting values of J+ and J-?)
7
Angular momentum, Spin and Scattering
1.5 EIGEN VECTORS OF J2 AND Jz
1.5.1 Total angular momentum
Total angular momentum is defined as sum of the orbital component and spin component
of angular momenta
J=L+S
J follows the same commutation rules as do L. if that is so S will also behave similarly.
Thus
[Jx,Jy] = iħJz ; [Jy,Jz] = iħJx ; [Jz,Jx] = iħJy
Example
Find [Jx, Jx]. Since [Jx, Jx] = [Jx, Jy] = Jx Jx,- JxJx it is identically zero.
We can show that Show that
[Jy, Jy] = 0 and so is [Jz,Jz]
Check your progress-2
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) Prove that J x J = iħJ
Hint – Write J = iJx + jJy + kJz, then find J x J and find if the sum of all
the three commutations produce the desired result.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ii) Prove that [Jx, J2] = 0
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1.5.2 Ladder operators
The operators
J+ (Jx + iJy)
(10)
8
Angular momentum
J- (Jx - iJy)
and
(11)
are called ladder operators. This will become clear in the following sections when we
operate those on appropriate Eigen functions.
Let us find properties of tehse operators.
1. What is [Jz,J+]?
[Jz,J+] = JzJ+,-J+Jz
= Jz(Jx + iJy) - (Jx + iJy) Jz
= JzJx + i Jz Jy – JxJz - iJyJz
= JzJx – JxJz + i (Jz Jy – JyJz)
= [Jz,Jx] + i[Jz,Jy]
= iħJy + i(-iħJx) = ħ(Jx + iJy)
= iħJ+
(13)
We can also Show that
[Jz,J-] = -ħJ-
(14)
and that J2 commutes with J+ and J-
What is the value of [J+,J-]?
[J+,J-] = J+,J- - J-J+
= (Jx +iJy) (Jx - iJy) - (Jx -iJy) (Jx +iJy)
= Jx2 + iJyJx – iJxJy +Jy2 +iJyJx – JxJy – Jy2
= i[Jy,Jx] + i[Jy,Jx] = 2i [Jy,Jx]
(15)
Some important cases
Case I
Consider
(,J2) = (,Jx2) + (,Jy2) + (, Jz2)
= (Jx, Jx) + (Jy, Jy) + (Jz, Jz)
(Jz, Jz)
Let us write (,m) then
((,m),J2(,m))
Or
i.e.
(Jz(,m), Jz(,m))
[(,m), ħ2J2(,m)] [ħm Jz(,m) , ħm Jz (,m)]
ħ2
ħ2m2
(16)
9
Angular momentum, Spin and Scattering
Case II
[Jz,J+] = (JzJ+ - J+ Jz) = ħ J+
Or
(JzJ+ ) = (J+ Jz) + ħ J+
Again writing (,m) we have
(JzJ+ ) (,m) = (J+ Jz) (,m) + ħ J+ (,m)
= J+mħ (,m) + ħ J+ (,m)
= ħ(m + 1) J+ (,m)
(17)
This equation suggests that the Eigen function J+ (,m) is an Eigen vector of Jz,
belonging to an Eigen value of (m + 1) ħ . It can be shown that J+ J+ (,m) is also an
Eigen vector of Jz with an Eigen value of (m + 2) ħ.
Important note- Eigen function and Eigen vector can be used interchangeably. Both
convey the same meaning. Similarly Eigen value is nothing but energy value.
Case III
[J2,J+] = J2J+ - J+ J2 = 0
Or
(J2,J+ )(,m) = (J+ J2)(,m)
= J+ ħ2(,m)
= ħ2 J+ (,m)
(18)
This equation suggests that J+ (, m) is an Eigen vector of J2 belonging to an Eigen
value of ħ2.
An Example
To find the maximum value of Jz
We note that J+ J+ J+ (, m) will have an Eigen value of (m+3) ħ for Jz and so on. Let
be the maximum Eigen value possible for a given value of ħ of J2, then
JzJ+(,m) = ( + 1) ħ J+(,m)
cannot be expected. Hence
JzJ+(,m) = 0
Or
J-J+(,m) = 0
But
J-J+(,m) = J2 - Jz2 - ħ Jz
Hence
(J2 - Jz2 - ħ Jz) (,m) = 0
Or
J2 (, m) - Jz2 (,m) - ħ Jz) (,m) = 0
10
Angular momentum
Or
(, m) - ħ22(,m) - ħ2 (,m) = 0
Or
( - ħ22 - ħ2) (,m) = 0
Or
( - ħ22 - ħ2) = 0
Whence
= - ½ + ½ (1 + 4) ½
(19)
Case IV
[Jz,J-] = JzJ- - J- Jz = - ħ JJ- Jz
- ħ J-
So
JzJ- =
and
JzJ-(,m) = J- Jz (,m) - ħ J-(,m)
= J- ħm (,m) - ħ J- (,m)
Or
JzJ-(,m) = (m-1) ħ J- (,m)
(20)
This equation indicates that J- (,m) is an eigen vector of Jz belonging to an Eigen
value of (m-1).
Reiterative Example
JzJ-(,{m-1}) = (m-2) ħ J- (,m)
It can further be shown that
the minimum value of Jz, belonging to J2 with the Eigen
value of ħ2 is
= - ½ - ½ (1 + 4) ½
(21)
An interesting task can be to Show that = - hence 2 must be an integer n.
Case V
What is ?
+ = 2 = n = 1 + (1 + 4) ½
Or
(n-1)2 = 1 + 4
Or
= (1/4)[(n-1)2 - 1] = [(n/2 – 1/2)2 – (1/2)2]
Or
=
n n
( 1) = J(J+1) where J = 0, 1/2, 3/2…
2 2
1.6 ORBITAL ANGULAR MOMENTUM AND SPHERICAL HARMONICS
1.6.1 Operators in polar form
To express various operators in polar form we will use following set of relations:
11
Angular momentum, Spin and Scattering
Set I
x = r sin cos
y = r sin sin
z = r cos
(22)
r2 = x2 + y2 + z2
(23)
tan2 = [x2 +y2]/z2
(24)
tan = y/x
(25)
Set II
Set III
Set IV
Now our task is to find
r r r
,
,
x y z
,
,
x y z
and
,
,
x y z
Once this is done their substitution in appropriate formulae would yield angular
momentum operator in polar form.
Step A
From
r2 = x2 + y2 + z2
2r
r
= 2x
x
Or
r
= x/r
x
Or
r
= r sin cos / r
x
r
= sin cos
x
It can be shown that
Or
(26)
12
Angular momentum
r
= sin sin
y
r
= cos
z
(27)
(28)
Step B
From
tan2 = [x2 +y2]/z2
we have
2 tan sec2 [/x] = 2x/z2
Or
x
= 2
x
z tan sec 2
Or
r sin cos
= 2
x
z cos 2 tan sec 2
Or
cos cos
=
x
r
(29)
cos sin
=
r
y
(30)
cos
=
z
r
(31)
Similarly
and
Step C
From
tan = y/x
we find that
sec2
= - y/x2
x
=
r sin sin
r cos 2 sin 2
r sin sin
r sec 2 cos 2 sin 2
Or
x
=
Or
x
=
2
2
cos ec sin
r
(32)
Proceeding in a similar manner we find that
13
Angular momentum, Spin and Scattering
and
How to find
cos cos ec
y
r
(33)
0
z
(34)
?
x
We know that x is a function of r, , and . In mathematical notation we express this fact
as x = x(r, , )
Recognizing this fact we can write
r
=
x
r x x x
Substituting values of
r
,
and
in this equation we get
x
x x
cos cos
cos ec sec tan
=
[sin cos ] +
[
]+
[
]
r
r
x r
Or
cos cos
= sin cos
+
r
x
r
cos ec sin
r
(35)
Students are expected to verify that
cos sin
cos
sin sin
y
r
r
r sin
sin
cos
z
r
r
and
(36)
(37)
1.6.2 Eigen values and Eigen functions of Lz
From
Lz i [ x
y ]
y
x
we have
Lz i [ x
y
]
y
x
14
Angular momentum
= i [ x(
r
r
) y(
)]
r y y y
r x x x
= i [r sin cos (
r sin sin (
Lz i
Or
cos sin cos cos ec
sin sin
)
r
r
r
sin cos cos cos sin cos ec
)]
r
1
r
r
It suggests that
Lz i
(38)
1.6.3 To find Lx operator
In a similar fashion we can find
Lx i [sin
cos cos ]
Ly i [ cos
and
(39)
cot sin ]
(40)
Once we have determined all components of L we can find the operator L (use L = i Lx
+ j Ly + k Lz ) and also another important operator L2 using the fact that
Or
Or
L2 = Lx 2 + Ly 2 + Lz 2
1 2
2
L2 = -ħ2 [ 2
(r
)]
r r
r
1
1 2
L2 = -ħ2 [
(sin
) 2
]
sin
sin 2
(41)
Important Note-In all the above text and text that follows we are using only h line or h
cross, which is equal to h (Planck’s constant) divided by 2. It is quite likely that print
does not show it distinctly. In some cases the size of h line is not same due to
unavoidable reasons but it conveys the same meaning.
Some conventions
In the above discussion we have written = (). It is more convenient to follow some
rules now onwards. Observe change in presentation that follows with use of for and
for .
15
Angular momentum, Spin and Scattering
Rule 1: A capital letter will denote function whereas a letter in lower case will represent
variable. Thus or more conveniently will represent a function of variable . This fact
can be expressed as () or ()
Rule 2: For brevity/convenience () will be written as .
Can now you suggest what would R(r) represent?
1.6.4 Eigen values and Eigen functions of L2
Let us write
L2 Y(,) = ħ2Y(,) ħ2Y
(42)
Using it for
L2 = ħ2 [
1
1 2
(sin
) 2
]
sin
sin 2
We obtain
1
Y
1 2Y
(sin ) 2
+ Y = 0
sin
sin 2
(43)
We can solve this equation by separation of variables writing
Y (,) = ()()
Substituting this in (23) we get
1
1 2
(sin
) 2
+ = 0
sin
sin 2
On multiplying it with
sin 2
sin 2
Or
we get
[
sin 2
1
1 2
(sin ) ]
=0
sin
2
[
1
1 2
= m2
(sin ) ]
2
sin
(44)
One simplest equation among these is
1 2
=
2
m2
(45)
This on simplification reads
16
Angular momentum
2
+
2
m2 = 0
(46)
Whose solution is
= Aeim
(47)
For the function to be single valued the necessary condition is
( +2) = ()
Which is equivalent to suggesting that ei2m = 1
This implies that m = 0; 1; 2; 3 etc.
What is the value of A?
Let us apply normalization condition to the wave function . The normalization condition
being
2
* d 0
0
2
2
A d 0
0
Or
A=
1
2
(48)
1 im
e
2
(49)
Thus the Eigen function for Lz is
=
Finding
In order to find we solve the second equation of the relation (24)
sin 2
[
1
(sin
) ] = m2
sin
Let us write cos = u and () = F (u) F
We get
dF
m2
[(1 u 2 )
] [
]F 0
u
du
1 u2
(50)
General case is clumsy to solve hence we take special case when m = 0.
17
Angular momentum, Spin and Scattering
dF
[(1 u 2 )
] F 0
u
du
(51)
To solve this equation we use power series for F(u) and write
F (u) =
dF
du
and
0
0
a r u r s
(r+s)a r u r s-1
dF
[(1 u 2 )
] [(1 u 2 ) 0 (r+s)a r u r s-1 ]
u
du
u
so that
[ 0 (r+s)a r u r s-1 ] [ 0 (r+s)a r u r s+1 ]
u
u
[0 (r+s)(r+s-1)a r u r s-2 ] [0 (r+s)(r+s+1)a r u r s ]
So substituting these values in relation (31) we get
0
[(r+s)(r+s-1)a r u r s-2 ] {(r s)(r s 1)a r u r s } a r u r s ] 0
(52)
Since this equation must be valid for all values of u, coefficients of all powers of u must
be independently zero.
Collecting the coefficients of ur+s in the above equation we get
(r+s+2)(r+s+1) ar+2 - (r+s)(r+s+1) ar +ar = 0
Or
ar+2 =
(r s)(r s 1 )
ar
(r s 2)(r s 1)
(53)
This suggests that the power series assumed for F would require its coefficients ar related
by above equation.
Check your progress-3
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i)
Find the values of a3; a4 and a4
Hint- coefficient a2 is the coefficient of u2, for which s = 0; a3 is coefficient of
u3 etc.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
18
Angular momentum
-------------------------------------------------------
But we can have a satisfactory solution only if power series is convergent, i.e., it
terminates after say arur, where r is some finite number.
It can happen if there occurs no term after ar ur . i.e.,
(r+s)(r+s+1) - = 0
Again taking simplest case when s = 0, we have
r(r+1) =
the solution of the equation with above condition is called Legendre polynomial Pl(u),
with the condition that Pl(1) = 1.
Thus
= B Pl (cos )
The constant B is again obtained from normalization condition for the wave function. Its
value is obtained as (should you do the labour of calculating it? Try)
2l 1 1/2
]
2
(54)
2l 1 1/2
] Pl (cos )
2
(55)
B= [
So that
[
and the Eigen function for L2 is
Y(,) =
1 im
e
2
[
2l 1 1/2
] Pl (cos )
2
(56)
In case m were not zero we find that
L2 Yl,m (,) = l(l+1) L2 Yl,m (,)
Where l(l+1) represents Eigen values of L2 and the corresponding Eigen function is Yl,m
(,)
Note that for a given value of l, the eigen value is (2l+1) fold degenerate. Since we can
associate m = -l; = - l+1; -l+2 etc. with it.
Note
19
Angular momentum, Spin and Scattering
When a function yields same Eigen value of more than one Eigen functions, the Eigen
functions are said to be degenerate
1.7 ANGULAR MOMENTUM AND ROTATIONS
An important fact: Angular momentum is related to rotation of a system
It is known that the transformation relations for a rotated frame about z-axis by an are
given by
x = x cos + y sin
y = y cos - x sin
z = z
(57)
when 0, we can write cos = 1 and sin = 0. Under this limiting conditions
x = x + y
y = y - x
and
z = z
(58)
1.7.1 Commutation relations for angular momentum operators
(1) To find [Lx , x]
Since Lx = (rxp)x = ypz – zpy here subscript suggests that the component along that
direction is considered. Thus py is component of p along y direction.
Writing Lx in operator form
L x = y (i
) z (i
)
z
y
But [Lx,x] = Lxx - xLx hence
[Lx,x] = { y (i
) z (i
) } x - x { y (i
) z (i
)}
z
y
z
y
[Lx,x] = - x { y (i
) z (i
)}
z
y
(59)
As the first bracket is zero because it is either differentiation of x with respect to z or with
respect to y.
What is [Lx,x] (x)?
20
Angular momentum
It is zero since (x) is function of x and not y or z.
Suggest what should be [Lx,x] (y)?
(2) Rule for [Lx,y]
We have [Lx,y] = { y (i
) z (i
) } y - y { y (i
) z (i
)}
z
y
z
y
Second term is meaningless. First term gives 0 + iħz
y
. Hence [Lx,y] = iħz.
y
It can be shown that
1. [Lx,z] = -iħy
2. [Lx, px] = 0
3. [Lx, py] = iħpz
(3) Let us find [Lx,Ly ]
[Lx,Ly ] = LxLy - Ly Lx
= (ypz-zpy)(zpx-xpz) – (zpx-xpz)(ypz-zpy)
= ypzzpx - zpy zpx - ypzxpz +zpxzpy+zpxzpy - xpzzpy
= y(-iħ) px - 0 -0 -0 -0 -0 -0 - x(-iħ) py
= (-iħ) ( ypx - x py ) = iħ (xpy - ypx)
= iħ Lz
(60)
Remember we have used the fact that perfect differentials along an axis do not operate on
other coordinates along the same axis viz. pzy = 0; pzx = 0 and pzz = - iħ, etc.
Proceeding in a similar fashion we can show that
1. [Ly, Lz] = -iħ Lx
2. [Lx, Lx] = 0
(4) To find commutation relation for [L2 ,Lx]
[L2 ,Lx] = [{[L2x+L2y+L2Lz]} ,Lx]
= [ L2x ,Lx] + [L2y ,Lx] + [L2z ,Lx]
Let us first compute [L2x ,Lx]
[L2x ,Lx] = Lx [Lx ,Lx] - [Lx, Lx]Lx
=0 + 0
Similarly we can show that [L2y ,Lx] and [L2z ,Lx] are also identically zero.
Hence [L2 ,Lx] = 0
21
Angular momentum, Spin and Scattering
Corollary
[Lx2, L2] = 0 and hence [L2,L2] commute.
The Hermitian operator
An operator is said to be Hermitian if it is self-conjugate. That is if A † = A . Thus for
Hermitian operator (A†,) = (,A).
Example
The operators J2, Jx, Jy, Jz are Hermitian. Try to verify.
Schwartz inequality
It says that a scalar product of a vector by itself is always greater than or equal to zero.
Let ( + ) denote a vector then
( + )* . ( + ) 0
(61)
Do you know?
In bra-ket notation it is written as <( + ) ( + ) > 0
Here is a number.
Note - We may also note that the product is done with the vectors after taking its
transpose (i.e., after changing row to column and vice versa). This is necessary as the
vector is represented as a column or a row matrix. Hence vector multiplication must
follow rules of matrix multiplication.
1.7.2 Eigen values and Eigen functions of Lz
Eigen function for Lz would be that which satisfies
Lz = c
= c
Or
-iħ
So that
= A eic/ħ
Since has to be single valued, if we replace with + 2, the wave function would not
change.
i.e.,
A eic/ħ
= A eic(+2)/ħ
Or
eic2/ħ
= 1
22
Angular momentum
or
2c/ħ
= 2m
Or
c =mħ
So that = A eic/ħ is its Eigen function with an Eigen value of m ħ.
Example
To find the Eigen value of (Lx iLy)
From
LyLz = i ħ Lx + LzLy
and
LzLy = - i ħ Lx + LzLy
we have
Lz ( Lx iLy) = Lz Lx iLzLy
= Lz Lx i [- i ħ Lx + LzLy]
= ħ (Lx iLy) + Lz(Lx iLy)
= (Lx iLy)(Lz ħ)
So
Lz ( Lx iLy) (m) = (Lx iLy)(Lz ħ) (m)
But
(Lz ħ) (m) = (m 1) ħ (m)
Hence
Lz ( Lx iLy) (m) = (m 1) ħ (Lx iLy) (m)
(62)
Thus if (m) is a wave function of Lz with an Eigen value of m ħ, then (Lx iLy) (m)
are also Eigen functions of Lz with Eigen values of (m 1) ħ.
Check your progress-4
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) What are the Eigen values of Lz for Eigen vector (Lx + iLy) (m)
and (Lx - iLy) (m)?
-----------------------------------------------------------------------------------------------------------------------------------------------------------
1.7.3 Eigen function and Eigen values for L2
Let represent the largest Eigen value for Lz when operated on (Lx + iLy) and its
maximum negative value for Lz when operated upon (Lx - iLy). Then
23
Angular momentum, Spin and Scattering
Lz(Lx + iLy) () = 0
and
Lz (Lx - iLy) () = 0
Consider
L2()
= (Lx2
+
Ly2 + Lz2) ()
= [Lz2 +(Lx - iLy) (Lx + iLy) – i(LxLy – Ly Lx)] ()
= [Lz2 +(Lx - iLy) (Lx + iLy) – i.i Lz] ()
= [2ħ2 - (Lx - iLy).0 + ħ] ()
Or
L2() = ( + 1) ħ2 ()
(63)
Similarly we can obtain
L2() = ( -1) ħ2 ()
(64)
Equations (53) and (54) must be valid at all the time hence
( + 1) must be equal to ( - 1)
Two possible situations would be
=
( + 1) =
and
We would disregard second solution as we have assumed that cannot have larger value
than . Hence we take
= ,
to be the acceptable result.
Let us call this value of as l. then the Eigen value of L2 are l(l+1) ħ2 .
What is the magnitude for the Eigen value of L?
We can claim that it must be square root of the Eigen value for L2.
Recall that quantum number related to L is written as l(l+1)
1.8 ROTATION OPERATOR
If Rz() represents an operator corresponding to an infinitesimal rotation about z-axis
then
Rz()(x,y,z) = (x,y,z)
= (x+y,y-x,z)
Now by Taylor expansion it can be written as
24
Angular momentum
= (x,y,z) +y
Rz()(x,y,z) = (1 + [y
Or
= [1 +
(x,y,z) - x (x,y,z)
x
y
-x
] )(x,y,z)
x
y
Lz] (x,y,z)
i
It suggests that
Rz() = [1 +
Lz]
i
(65)
And similarly we can show that
and
Rx() = [1 +
Lx]
i
Ry() = [1 +
Ly]
i
(66)
In other words Lx, Ly and Lz are the generator of rotations about z-axis.
1.8 ROTATIONAL INVARIANCE AND CONSERVATION OF ANGULAR
MOMENTUM
Invariance dictates no change under some operation, and conservation dictates
commutation with energy operator. Rotational invariance is the property of a system
such that after undergoing rotation, the new system still obeys Schrodinger equation.
Thus for any rotation R, the rotation operator and energy operator commutes.
i.e. [R, (E-H)] = 0
Since rotation does not depend explicitly on the time, it commutes with energy operator.
Thus [R, H] = 0 here H represents energy operator or the Hamiltonian of the system.
Let the system be rotated (say in x-y plane) by an infinitesimal angle d then the rotation
operator is
R = 1 + Jz d
Then from [R, (E-H)] = 0 we find that [(1 + Jz d ),
Or
d
(Jz) = 0
dt
d
] =0
dt
(67)
25
Angular momentum, Spin and Scattering
It suggests that angular momentum in such rotations is conserved.
1.10. ROTATIONAL DEGENERACY
Let us have a further infinitesimal rotation of , then we will have
Rz ( +) = Rz(). Rz()
= [1 +
Lx] Rz()
i
dR z ( )
lim. R z ( ) R z ( )
i
Lz
0
To give
Rz () = e-iLz/ħ
(68)
Example
To find a rotation in spherical polar coordinate system
For this we will put r = r; = ; and = - so that
Rz()(r,,) = (r,,-)
Or
[1 - i
Lz] (r,,) = (r,,) -
i
Comparing the two sides we obtain
Lz = - i
Degenerate functions
Let us find out what happens if rotation about z-axis totals 2? This new state is
indistinguishable from the state with no rotation. Such state then belongs to the same
energy, representing degenerate functions.
1.11 LET US SUM UP
Angular momentum is one of the basic properties of nature. Among six universal
conservation laws, it has lot of significance in physical world. Essentially we have
two kinds of angular momenta namely orbital or rotational angular momentum
26
Angular momentum
denoted by L and other is spin angular momentum denoted by S. The total of the
two LS is denoted by J. for a physical system it has to be positive. J also
determines total energy of the system.
Quantum mechanics being the most reliable description of nature we apply it in
solving the physical problems. Solving physical problems entails finding an
appropriate Eigen function for the system and then solving it for finding Eigen
values related to those functions.
Solving physical problems easy when we use angular momentum in operator
form. The mathematic of operators is facilitated by commutation rules. Increase in
energy (excitation) and decrease in energy (de-excitation) can be computed by so
called ladder operators.
Rotation of frame of reference, which is equivalent to change in angular
momentum.
1.12 CHECK YOUR PROGRESS: THE KEY
1.
i) [p,x] = [(ipx + jpy + kpz), x]
= [ipx,x] + [jpy,x] + [kpz,x]
= iħ + 0+ 0
ii) (x2, px) = x(x, px) + (x, px) x. Substitute values of commutation brackets
namely [ipx,x] = iħ and (x, px) = - iħ for result.
2.
i) J x J = i(JyJz-JzJy) + j(JzJx-JxJz) + k(JxJy-JyJx)
= i[Jy,Jz] + j[Jz,Jx] + k[Jx,Jy]
= i(iħJx) + j(iħJy) + k(iħJz)
= iħ(iJx + jJy + kJz) = iħJ. Remember i denotes square root of minus one
whereas i represents unit vector along x-axis.
ii) [Jx, J2] = [Jx,(Jx2 + Jy2 +Jz2] = [Jx,Jx2] + [Jx,Jy2] + [Jx, Jz2] = 0
3.
From ar+2 =
(r s)(r s 1 )
ar
(r s 2)(r s 1)
for a3 put s=0 and r=1to get (2-)/6
similarly a4 = 2(3-)/12 etc.
5. The values are (m+1) ħ and (m-1)ħ by using equation 62.
27
Angular momentum, Spin and Scattering
UNIT 2 SPIN
Structure
2.0 Introduction
2.1 Objectives
2.2 Electron Spin
2.3 Spin ½ and Pauli matrices
2.3.1
Spin and spin operator
2.3.2
The spin matrices
2.3.3
The Pauli spin matrices
2.4 Observable and Wave functions of Spin ½ particles
2.4.1
Spin matrices and Eigen functions
2.4.2
Electron spin functions
2.4.3
Electron spin functions
2.4.4
Further Pauli spin matrices
2.5 Spin of Fields
2.6 Vector Fields and Particles of spin 1
2.6.1
Vector fields and particles
2.6.2
Spin matrices as operators
2.7 Spin Independent Interactions of Atoms
2.8 Spin Independent Nucleon-Nucleon Interactions
2.8.1 Isospin
2.9 Let Us Sum Up
2.10 Check Your Progress: The Key
2.0 INTRODUCTION
28
Angular momentum
To trace the history of an intensive study of any topic in Physics we find that it was the
spectrum of hydrogen atom. Hydrogen atom consists of a proton and an electron located
near to it in some permitted orbit. The first excited state lies about 10.4 eV above the
ground state. But it was found that even the ground state was not singlet (?). In order to
explain this, electrons and protons were assumed to possess spin. This spin was assumed
to be responsible for hyperfine splitting of energy levels. Spin states were to be either
spin up or spin down to accommodate experimental data. With the development of
quantum mechanics various aspects of spin were studied. This unit will enable students to
appreciate spin functions, spin matrices and discussion of a few spin half systems of
particles.
2.1 OBJECTIVES
This learning material is intended to acquaint the students with
i) Concept of spin and spin operators
ii) Spin Eigen functions and Eigen values
iii) Pauli spin matrices
iv) Spin operators
v) Some examples related to spin of particles
2.2 ELECTRON SPIN
2.2.1 Need to have a spin
If electron were a spin less particle and only described an orbital motion about the
nucleus, its magnetic dipole moment would have been
e
L
2m 0
where m0 is the mass of the electron. This magnetic moment then should be the source of
permanent magnetism of the ferromagnetic substance. But the observed magnetic
moment of a magnetic specimen gives a factor
e
e
and not
. This has been
m0
2m 0
attributed partially to gyro magnetic effect. Further for S (l= 0) state atom should have no
magnetism. This is again against the observed fact.
Fine structure of the spectrum is also inconsistent with the orbital angular momentum
concept. Sodium D lines are well known example. Fine structure is possible only if there
29
Angular momentum, Spin and Scattering
were additional energy levels. Zeeman and Paschen-Back effects too cannot be explained
without attributing spin to electron.
For S state there should be no splitting of atomic beam in Stern-Gerlach experiment, but
this also goes against the experimental observations.
To explain all above anomalies Uhlenback and Goudsmit postulated that electron
possesses an intrinsic angular momentum. He named it electron spin, which for brevity
we simply call spin. Ultimately it was noted that protons and even neutrons possess spin.
2.3 SPIN ½ AND PAULI MATRICES
2.3.1 Spin and spin operator
The spin associated with an electron is quantized. For explaining doublet fine structure of
alkali atoms it was sufficient to ascribe a spin of ½ ħ to the electron. Associated spin
quantum number, s has only one value for the electron namely ½. We describe the
quantum state of electron by ħ. Assuming to be related to quantum number s, the
multiplicity of energy levels would be (2s + 1). According to Stern-Gerlach the atomic
beam with S = 0, splits into two beams, i.e., the multiplicity is 2. It suggests that s = ½.
Note- S is total spin magnetic quantum number and is given by S = s1 s2 s3 ..Where
s1,s2,s2, are spins component electrons of the system or the state. Incidentally all s have
the same value of ½.
Remember whenever we talk of angular momentum we add ħ with the quantum number.
It is also usual to write angular momentum in units of ħ. In that case again ħ is omitted.
Thus we conclude that each electron in a given orbit provides a spin contribution of ½ ħ
to the angular momentum.
Spin operator S represents spin angular momentum operator. Its components are 3x3
spin matrices.
0 0 0
Sx i 0 0 1
0 1 0
0 0 1
Sy i 0 0 0
1 0 0
0 1 0
and
(1)
Sz i 1 0 0
0 0 0
30
Angular momentum
S commutes with L but not with Hamiltonian H. We must note here that (L + S) together
commute with H. and hence (L+S) must be a constant of motion not L or S
independently.
Let us find S2 from above relation
We know that
S2 = Sx2 + Sy2 + Sz2
Substituting matrix values in the above we get S2= 2ħ2.
It may be noted that in general L2 is not a constant of motion. But S2 is constant of motion
and commutes with all the dynamical variables. While computing the energy values S2
can be replaced by s(s+1) ħ2. Here s is called an intrinsic spin angular momentum whose
value is ½.
Important Note- in this and following discussions we have used ħ, h line (or h cross)
everywhere whose value is Planck’s constant divided by 2. It is quite likely due to
prints in various fonts it may not become explicitly clear.
2.3.2 The spin matrices
(sx), ( sy) and (sz) are called spin matrices and in the matrix form are represented as
0 1 0
sx
1 0 1
2
0 1 0
0 i 0
sy
i 0 i
2
0 i 0
1 0 0
(2)
sz
0 0 0
2
0 0 1
1 0 0
2
2
So that
s 2 0 1 0
(3)
0 0 1
The necessary condition is that S is not expressed in terms of r and p.
Examples
Mesons are spin zero; electrons, protons, neutrons, neutrinos and mesons are spin
half; and photons and gravitons are spin one particles. Try to find what spin a graviton
possesses.
If we place a spin half particle in a uniform magnetic field B for a time , its probability
amplitude becomes e
i zB
times what it would be in no field. Here z is either + or – of
some number. If the particle is not in pure spin up or spin down state we can describe its
31
Angular momentum, Spin and Scattering
condition in terms of its amplitude to be both in the pure up or pure down state. But in a
magnetic field those states will have phases changing at different rates. Hence on what
time and how long the particle is in the field matters most.
Aggregates of particles that are sufficiently tightly bound can be regarded as a composite
particle. It can then be characterized by definite magnitudes of their total internal
momentum as long as their internal motions and the relative spin orientations of their
component particles are not significantly affected by the interactions between the
aggregates. The spin of the aggregate is found thus by observing the triangle rule of
vector addition: the magnitude of the sum of two angular momentum vectors can have
any value ranging from the sum of their magnitudes to the difference of their magnitude,
by integer steps. It also indicates that the sum of the z-component of the angular
momentum equals that of their resultant. We can show this by applying it to linear
combination of products of Eigen states of two commuting angular momentum operators
that are Eigen states of total angular momentum. The same kind of addition formula
holds for products of rotation or tensor operators and for the states produced when a
tensor operator acts on an angular momentum Eigen state.
Thus the spin of aggregates (of n particles) each of which has spin half plus any number
of particles with spin zero, ignoring internal angular momentum of these particles. The
value of s can be from 0 to ½ n. Total orbital angular momentum however can be zero or
some integer in general. It can be a good exercise to see what kind of value results if spin
and orbital angular momenta are added together.
Aggregate of particles that have zero or integral spin are described by symmetric wave
functions. Such an aggregate obeys Bose-Einstein statistics. Particles that have half or
half of odd integral spins are described by anti-symmetric wave functions. Such particles
obey Fermi-Dirac statistics.
2.3.3 The Pauli spin matrices
The two state system of electron spin is very useful and hence is conveniently represented
as sigma matrices. Anybody working in quantum mechanics needs to memorize these
popularly known as Pauli spin matrices.
0 1
0 i
1 0
x =
(4)
; y =
; z =
1 0
0 1
i 0
Product of these matrices taken two at a time also finds use in the analysis of interactions.
The three matrices are further analogous to the three components of a vector called sigma
() vector
2.4 OBSERVABLE AND WAVE FUNCTIONS OF SPIN ½ PARTICLES
32
Angular momentum
2.4.1 Spin matrices and Eigen functions
The spin coordinates take on 2s + 1 values for a particle of spin s. A convenient set of
ortho-normal one particle spin functions is provided by the normalized Eigen functions of
the J2 and Jz matrices. These Eigen functions are (2s + 1) row and one-column matrices
that have all elements zero except one.
Example
What are the spin functions for s = 3/2?
The four spin Eigen functions are
1
0
0
0
0
1
0
0
u (3 / 2)
; u (1/ 2) =
; u ( 1 / 2) =
; u ( 3 / 2) =
(5)
0
0
1
0
0
0
0
1
The corresponding Eigen values are 3ħ/2, ħ/2, -ħ/1 and -3ħ/2. Here we have used double
lines to indicate that these denote matrices and not determinants.
From simple matrix-matrix multiplication rule one can verify the orthonormality by
finding that u1*u1 = 1; u1*u2 = 0 etc.
It can be shown that thee are a total of (s+1)(2s+1) symmetric and 2(2s+1) antisymmetric states for two-particle system with same spin.
Check your progress-1
Note: a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
iii) What will be u2* u2 and u2*u4? Assume u1 u(3/2), u2 u(1/2),….
--------------------------------------------------------------------------------------------------------------------------------------------------iv) How many anti-symmetric spin states are possible with above four
u functions?
--------------------------------------------------------------------------------------------------------------------------------------------------2.4.2 Electron spin functions
For brevity the electron spin matrices can be written as S = ½ ħ where
0 1
0 i
1 0
x =
; y =
; z =
1 0
0 1
i 0
These matrices are called Pauli spin matrices. For these matrices the normalized Eigen
functions are written as
1
0
u(1/2) =
;
u(-1/2) =
(6)
0
1
33
Angular momentum, Spin and Scattering
these have Eigen value of ½ ħ and -½ ħ respectively.
As a corollary it can be shown that the Pauli matrices are both Eigen functions of S2 and
have same Eigen value of ¾ ħ2.
2.4.3 Electron spin functions
The spin matrices can be written as S = ½ ħ, where are given be equation (5). The
normalized Eigen functions of S
u(1/2) =
1
0
;
u(-1/2) =
0
1
are abbreviated as (+) and (-) respectively. So hereafter when we write (+), it would
imply that we are concerned with u(1/2) state and when we write (-) it is for u(-1/2) state
with Eigen values of ½ ħ and -½ ħ respectively.
If we take two electrons system we can have four independent wave functions, viz.,
(+ +), (+ -), (- +) and (- -).
These Eigen functions can be regrouped in a combination that are Eigen functions of
(S1 + S2)2 and S1z and S2z.
Check your progress-2
Note: a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) xy = - yx = i.
--------------------------------------------------------------------------------------------------------------------------------------------------ii) x(+) = (-); y(+) = i(-) ; z(+) =(+); x(-) = (+) ; y(-) = -i(+) ;
z(-) = -(-).
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------iii)How would you verify the orthogonality of wave functions for two
electrons?
--------------------------------------------------------------------------------------------------------------------------------------------------iv) How would you write Pauli spin matrix as a 4x4 matrix?
--------------------------------------------------------------------------------------------------------------------------------------------------2.4.4 Further Pauli spin matrices
34
Angular momentum
The two state system of the electron spin is so important that it is very useful to have a
neat mode of presenting them.
We know that the Hamiltonian for the electron in a magnetic field is
Hij = - ( ijBx + ij By + ij Bz)
If we take the magnetic field along z-direction, then
(7)
11 = 1; 12 = 0; 21 = 0; and 22 = -1
This gives
H12
H
Hij = 11
H 21 H 22
For electron in a magnetic field Bz we have
Bz
( Bx iBy )
Hij =
Bz
( Bx iBy )
Terms for x-direction are found to be
(8)
(9)
11 = 0; 12 = 1; 21 = 1; and 22 = 0
In shorthand notations ’s are represented as in equation (5).
Anyone pursuing quantum mechanics need to memorize those matrices.
2.5 SPIN OF FIELDS
The spins of electrons and protons are considered to be connected with their moments of
motion. Spin is considered to be just a quantum mechanical quantity. Yet we can assume
that spin has some physical significance. If in a substance the spins of particles have a
preferred direction, then this is interpreted as spin polarization of the substance. Every
substance creates a spin field in the space surrounding it when polarized by spins. (This
field is also called ‘axion field’).
Let us examine this fact: the electric charge manifests as the electric field in the space
surrounding it; the magnetic moment manifests itself as magnetic field. From the analogy
we can expect that the manifestation of spin result in a hypothetical spin field.
The spin of elementary particles can be considered as a source for the spin field. It can
safely be assumed that a spin field is produced by spin polarization, i.e., selective
orientation of spins in space. The simplest way to achieve a selective spin orientation is
through the mechanical rotation of the objects. On such rotations spins are oriented along
the axis of rotation. You can seek the details of such a spin field generators from different
websites.
2.6 VECTOR FIELDS AND PARTICLES OF SPIN 1
35
Angular momentum, Spin and Scattering
A field is a physical quantity, which takes on different values at different points in space.
In other words field is a mathematical function of position and time. For vector fields we
can visualize by drawing various vectors at various points in space, each of which gives
the field strength and direction of the field at that point.
Vector field is characterized by some amount of a physical quantity called flux, coming
in or going out from the space.
2.6.1 Vector fields and particles
In the study of elementary particles we also discuss a small family of particles that do not
come under leptons, mesons or baryons. These are field particles that are responsible for
carrying the forces in order to enable particle interactions.
We now believe that inter particle interaction is essentially through the field they
establish (the concept of action at a distance was thus replaced by the notion of a field).
According to it one particle sets up a field around it and the other particle interacts with
the field when it comes under its influence.
The quantum field theory further modifies this notion by announcing that the fields are
carried by a quantum related to that field. In this point of view first particle emits a
quantum the second particle responds to the field then by absorbing the quantum. A force
accomplished through the exchange of particles is called an exchange force.
Check your progress-3
Note: a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) A system of elementary particles was found to have all spin
magnetic moments along z-axis. What kind of property must be
associated with the space about this system?
--------------------------------------------------------------------------------------------------------------------------------------------------ii) If action at a distance were the central idea for field concept, list at
least two fields known to you.
-------------------------------------------------------------------------The particles responsible for some known fields are summarized below.
Interaction force
Field particle
symbol
charge
Gravitation
graviton
0
spin
2
36
Angular momentum
Weak
Electromagnetic
Strong
weak bosons
photons
gluon
W+,W-,Z0
g
1, 0
0
0
1
1
1
Can you tell which force/s among those listed in the table do not produce a vector field?
2.6.2 Spin matrices as operators
We have suggested that a column matrix and a row matrix well represent a vector. Hence
we will discuss about matrices associated with the spins to get further insight into this
phenomenon.
Let us find z + >
By multiplying it with < + ,
we get
< + z + > = 11 = 1
Now let us see the effect of multiplying it by < - from left
< - z + > = 21 = 0
From above two equations we get a clue that if we have a relations like
< + ? > = 1 and < - ? > = 0
There can be only one state vector + > that satisfies both the relations. It suggests that
z + > = + >
(10)
Example
To find y z + >
We have y z + > = x (i ->)
= i x ( ->)
=i + >
Properties of operators
x ( +>) = - ->
x ( ->) = +>
y + > = i - >
y - > - i - >
z + > = + >
37
Angular momentum, Spin and Scattering
z - > = - - >
(11)
One can easily show that Prove that x y = i z
2.7 SPIN INDEPENDENT INTERACTIONS
Collision of identical particles each one having (2s+1) Eigen functions produce (2s+1)2
independent spin functions for each of the pairs. Any of (2s+1)2 linearly independent
combinations of these products can be used to describe them. However they are divided
in to three classes.
First class consists of one particle spin function in which both particles are in the same
spin state. Example: u1.u1 or u2.u2 etc.
Second class consists of sum of the products. Obviously these two classes correspond to
symmetric functions in which interchange of spins between two particles has no effect on
the system. Example: u1.u2 or u3.u4.
The third class has differences of products. It produces an anti-symmetric state.
Example: u1.u4, u2.u3.
2.8 SPIN INDEPENDENT NUCLEON-NUCLEON INTERACTIONS
2.8.1 Isospin
Neutrons and protons in a nucleus possess some mass and are bound in the nucleus with a
strong force. They differ in that – one is charged and the other is a neutral particle.
Heisenberg suggested that like an electron, a nucleon also possesses spin half magnetic
moment. Spin half nucleons also then occupy two states: the isotopic spin up (protons)
and isotopic spin down (neutrons) states. Note that just like electron states cannot be
identified in the absence of magnetic field (spin up or spin down) so is the case with
nucleons. We cannot assess in which spin state a nucleon lies in the absence of a
magnetic field.
Nucleon is an isospin particle with nuclear spin I = ½ ħ., the Eigen values for protons and
neutrons are + ½ and – ½ respectively.
The Pauli matrices for nucleons are represented as I = ½ . Those are given below0 1
0 i
1 0
1 =
; 2 =
; 3 =
1 0
0 1
i 0
(12)
These also follow commutation relations as shown below[1 , 2 ] = 2i3
etc.
(13)
38
Angular momentum
We can represent a nucleon as (nucleon) > =
p
n
= p
1
0
+ n
0
1
(14)
Example
1
0
refers to proton and ½ -½ > =
refers to a neutron.
0
1
Can you suggest that if the Eigen function for isospin are Iml> than Iz Iml> = ml Iml>
1 0
Hint : Iz = 3 =
,operate Eigen function with it. Alternatively in bra-ket
0 1
½½>=
notation the Eigen value ml is contained in the ket.
2.9 LET US SUM UP
For long the fine structure of spectral lines was debated. With the availability of
more and more sophisticated instruments fine structure was established beyond
doubt. In the initial period there was no successful and plausible explanation.
Uhlenback and Goudsmit tried with a solution giving a revolutionary idea without
proof that electrons possessed spin. They claimed that like any celestial body
electron also spin about its own axis. Though it is still under debate whether
electron spins, yet has been established beyond doubt that electron possesses a
spin magnetic moment that is responsible for the splitting of energy levels.
Once the concept of spin was established all treatment of new theory was obvious.
In the effort to find functions and energy values electron spin functions and
related matrices were defined. The most important result was the explanation of
action at a distance.
The essentials of action at a distance lied in the spin field that enabled to have
field particle interaction. Iso-spin concept enabled to find functions and energy
levels for collection of Fermi particles.
2.10 CHECK YOUR PROGRESS :THE KEY
39
Angular momentum, Spin and Scattering
1. i) Substitution of u2* and u2 gives u2* u2 = 1 and similarly substritution of u2*
and u4 gives u2*u4 = 0
ii) For four functions: symmetric functions are u1u1, u2u2, u3u3, u4u4, u1u2, u3 u4.
Other combinations produce anti-symmetric namely u1u3, u1u2-u2u3, etc.
0 1
2. i) x y = i z is verified by multiplying x matrix
with y matrix
1 0
0 i
1 0
. Which is z matrix multiplied by i.
. The resultant matrix is i.
0 1
i 0
Similarly changing the order verifies the other part.
0
1
0 1 1
0 1 0
ii) x. (+) =
0 = 1 similarly x. (-) =
1 = 0 ,
1 0
1 0
other identities are also verified by applying matrix multiplication rules for
corresponding matrices and identifying the resulting matrix.
iii) Orthonormality of function is verified by showing that when one function in
matrix form multiplied by complex conjugate of other matrix, the result is a null
matrix.
0
iv) Pauli matrix as 4x4 matrix is written as
where represent simple
0
0 0
2x2 matrix and 0 represents 2x2 null matrix
.
0 0
3. i) When magnetic moments are aligned it says all spins of the particles are
aligned.
ii) Gravitational and Coulombian fields are examples of action at a distance.
40
Angular momentum
UNIT 3 SCATTERING THEORY
Structure
3.0 Introduction
3.1 Objectives
3.2 Definition of Scattering Cross Section
3.2.1 Scattering
3.2.2 Scattering Cross Section
3.3 Stationary Wave Scattering
3.3.1 Collision of identical particles
3.4 Representation of Scattering phenomenon by a bundle of Wave packets
3.5 Scattering of a Wave packet by a Potential
3.6 Calculation of Cross section
3.7 Laboratory system and Centre of Mass system
3.7.1 Connection between center of mass and laboratory system
3.7.2 Relation between scattering angle in the two coordinate systems
3.8 Scattering by a Controlled potential wave: Analysis and Phase Shift method
3.8.1 Scattering by a square-well potential
3.8.2 Phase shift in scattering by a spherically symmetric potential
3.9
Impact Parameter
3.10 Relation between Phase shift and Logarithmic derivative
3.10.1 Effect of spin
3.11 Behaviour of Phase Shift at Low energies
3.11.1 Phase shifts at low energies
3.12 Scattering by a Hard Sphere
3.13 Let Us Sum Up
3.14 Check Your Progress: The Key
41
Angular momentum, Spin and Scattering
3.0 INTRODUCTION
When a stream of particles (called projectiles) strike randomly oriented objects then an
incident individual particle is sent in any of the all-possible directions, by the randomly
oriented objects (called target). Collectively speaking different particles of incident beam
are sent in different directions. However the sum of particles sent by objects of the target
is same as the number of incident particles. But in the process of sending projectiles back
in different direction (usually called scattering), the phases of particles also undergo a
change. The phase change matters most when we discuss situations in microscopic
domains. To understand the scattering we first undertake two-particle collision and then
develop a consistent theory for the beam of the particles. Ultimately we will take up
different systems for scattering with fully quantum mechanical aspect. It is only after the
advent of quantum mechanics scattering of micro-particles (like alpha and other sub
atomic particle), the probe into the atoms and even nucleus was that led to the better
understanding of the matter. Since the properties of the scatterer are reflected in the
scattering process scattering experiments has become an important method now for the
study of atoms, molecules, nuclei and fundamental particles.
3.1 OBJECTIVES
After studying this unit students will be able to
i) Understand two body dynamical problems
ii) Understand scattering process
iii) Know the factors on which scattered beam intensity depends and hence
will be able to estimate the scattering cross section
iv) Recall scattering under different conditions
v) Find relationship between wave associated with projectile and size of the
scattering atoms.
3.2 DEFINITION OF SCATTERING CROSS-SECTION
3.2.1 Scattering
42
Angular momentum
Scattering is a phenomenon that takes place when a collimated beam of particles with
well defined energies and properties (called projectiles), collide a target of atoms and
ultimately shatter in all possible directions. When we talk of miniscule matter (refers to
electrons, atoms, nuclei and so on) moving with relativistic speed we dare to associate
with it a wave. In scattering, wave functions associated with the particles do not vanish at
large distances from the scatterer. In short we consider only non-relativistic potential
scattering of particles.
Scattering experiments are the standard means of investigation in to structures and interparticle interactions in atomic, nuclear and particle physics. The energy of the beam and
the nature of the particles in it determine the extent of scattering in different directions.
The solutions might also correspond to the scattering by a force field, where energy is
specified in advance and the behaviour of wave functions is found in terms of energy.
The measurements indicate that different number of particle appears in different
directions.
3.2.2 Scattering cross section
The scattering measurements involve intensity of the beam as a function of angle (or may
be some other property of the projectile – target system). Most common measurements
are done on cross section. To avoid complexities, we assume that the wavelength
associated with the incident beam of particles is small (as is the case with matter waves)
as compared to inter atomic distances of the target. Further we will analyze scattering by
a single atom or nucleus and then add the separate effects of various scattering centers.
Consider a beam of particles moving along z-direction with I0 particles per unit area per
unit time. When particles undergoes scattering then the number of particles received by
the detector per unit time is given by
dN = () n I0 d
(1)
where n gives the number of scatterers in the target.
() is numerically equal to the cross sectional area of the projectiles scattered per
scatterer of the target into a unit solid angle in the direction of . Essentially this ratio
43
Angular momentum, Spin and Scattering
has the dimensions of an area, and so it is called a cross section. (However, it is pertinent
to notice that this is not basically an area, but only a probability). For this treason ()
is called the differential scattering cross section. If we integrate differential scattering
cross section over the solid angle we get total scattering cross section. When the particles
undergo scattering through an angle at an azimuthal angle of , due to a potential, then
the angular distributions of particles is related through
(,) d =
(Numberof particles scattered into a solid angle d per unit time)
I0
(2)
Where (,) is scattering cross section and d is the differential solid angle along
(,). See Fig.1
For spherically symmetric potentials (,) is independent of . The cross section of
scattering for a spherically symmetric potential is given by
d
Fig. 1
=
2
0
0
[ ( ) sin ]d
d
= 2 [ ( ) sin ]d
(3)
0
In the above analysis we assume that the scattering centers (scatterers) act independently
of each other. Further this expression holds well for a wave as for a beam of particles.
Physical interpretation
44
Angular momentum
Assume that we have a target, which has N scatterers per unit volume. A beam of
particles of cross section A and intensity I0 strikes normally on it. Then the intensity I (x)
of the beam after penetrating a distance x into the target is related to the loss of intensity
dI(x) on penetration of dx into the target. The relation is given by
-
A dI(x) =
A
NA dx
(4)
[Note that (Na dx) is the number of scatterers in volume (A dx). Here we have ignored
multiple scattering].
On integration we find that
Or
dI ( x)
NA dx
I ( x) A
I(x) = I0 eNx
(5)
Here N represents macroscopic cross section and is interpreted as inverse of mean free
path.
3.3 STATIONARY WAVE SCATTERING
The Schrodinger equation for scattering can be written as
{(ħ2/2)2 + V(r)} = E(r)
Where E is related to p (= ħk), the momentum of homogeneous particle wave proceeding
in positive z-direction, through E = (p2/2). The wave is given by
(r) = (z) = eikz
3.3.1 Collision of identical particles
Consider a collision of two particles a and b. After collision particle a goes along path1
and particle b goes along path 2 as shown in Fig.2.
If amplitude of happening of this event is f () then the probability P1 of observing it is
proportional to [f ( )] 2. If however the particle a goes along 2 and particle b goes along
1, the probability of occurring this process is P2 = [f (-)] 2 , see Fig.3.
45
Angular momentum, Spin and Scattering
1
a
b
2
Fig.2
Ignoring the spins of the individual particles and assuming particles a and b to be
identical, we will have amplitude that any one particle goes along 1 and the other goes
along 2 given by
f () + f ( - )
1
a
2
b
Fig.3
In collision as also in scattering, the forces of mutual interaction acting on two particles
need also to be considered.
It is usually convenient to consider these problems in center of mass coordinates, so that
the relative position vector r = r2 – r1 continuously changes.
The position of center of mass is
rcm =
m1r1 m2 r2
m1 m2
(6)
If both the particles are of the same mass i.e., m1 = m2 we have
r r
rcm = 1 2
2
The scattering function is given by
1
u (r ) r eikz f ( , )eikr
r
Where r, , are the coordinates of the relative position vector r.
The asymptotic forms of wave functions can be written as
(7)
(8)
eikz eikz [ f ( , )] f ( , )
eikr
r
(9a)
eikz eikz [ f ( , )] f ( , )
eikr
r
(9b)
and
46
Angular momentum
The differential scattering cross section is given by
( , ) [ f ( , )] f ( , )]2
[ f ( , )]2 [ f ( , )]2 2 Re f ( , ) f *( , )
When the particles are indistinguishable the interference term is zero and so
( , ) [ f ( , )]2 [ f ( , )]2
(10)
Which combines the probabilities of observing incident and scattered particle.
Particles, which interfere with positive sign, are bosons while those interfere with
negative sign are fermions.
Check your progress-1
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) Which is the interference term in equation above equation (10)?
-------------------------------------------------------------------------ii) Write equation (10) for electron.
-----------------------------------------------------------------------
3.4 REPRESENTATION OF SCATTERING PHENOMENON BY A BUNDLE
OF WAVE PACKETS
The scattering problem is a very difficult problem if the coordinates of two particles are
time-dependent. Such that they are initially in the form of separated wave packets, then
they move into the common region. There they interact, and finally separate into two
wave packets moving in a direction that has different probability depending on several
factors. This problem can be handled safely if we consider it as a time-independent,
stationary-state problem that is much easier to solve. In the time-independent, stationarystate problem we consider the state of one particle as corresponding to a plane wave at
large distances. The other particle (target) is assumed to be stationary. The plane wave
interacts with a fixed potential of the other particle. Outgoing waves are identified with
47
Angular momentum, Spin and Scattering
the scattered particle. The process when conceived to be a steady one can then very well
be represented by the usual interpretations of the wave function. It is usual practice to
describe it in the center of mass system, in which a particle of mass interacts with the
field of a potential of another particle at the origin. Try to answer ‘What is mean by
isotropic scattering?’
3.5 SCATTERING OF A WAVE PACKET BY A POTENTIAL
Strictly speaking scattering problem needs temporal* description (*temporal means as a
function of time). But when there is a steady current of particles approaching a target
from a very large distance we can use time independent Schrodinger equation.
2(r) + (2/ ħ2 ) [E – V(r)] (r) = 0
where is the mass of the particle in the beam.
For more than one particle is the reduced mass and E is the energy in the center of mass
system.
Let us consider a plane incident wave and an out going scattered wave. Then the solution
of the Schrodinger equation has the form
(r )
lim. ikr
eikr
e f ( )[ ]
r
r
(11)
where k represents vector along z-axis and the potential is assumed to be spherically
symmetric. The first term represents wave incident on the scatterer (scattering centre).
Further, we have assumed that the amplitude of the scattered wave is independent of the
azimuthal angle .
If the incident beam is along z-direction, the equation (12) becomes
(r )
lim. ikz
eikr
e f ( )[ ]
r
r
The flux density is given by the second term. If is proportionality term depending on
angle then the scattered flux and incident wave flux are given respectively by
eikz
f ( )[ ]
r
2
and
eikr
f ( )[ ]
r
2
(12)
respectively.
48
Angular momentum
Let the detector be kept at a distance r and the area dA intercepts those waves, then the
flux
eikr
f ( )[ ]
r
Or
eikr
f ( )[ ]
r
2
dA = f ( )[eikr ]
2
dA
r2
2
2
= f ( )[eikr ] d
dA
(13)
Equation (13) represents the number of waves passing through the detector of area dA.
Here we have written
dA
= d, the solid angle subtended by the detector at the scatterer.
r2
Thus the differential scattering cross section is
2
( )d
f ( )[eikr ] d
f ( ) d
2
2
= [eikr ] d
(14)
3.6 CALCULATION OF CROSS SECTION
= R2 is the general formula for the cross section of a spherical body. But it is only a
geometrical cross section. We notice that the cross section defined for scattering does not
match with it. The reasons are several. A few of them are- the scattering occurs from the
interaction of particle with the potential function of the scatterer which may be different
for particles of different natures, energy of the projectile causes the projectile to penetrate
the field of scatterer through different extent, the scattering centers may be
inhomogeneously distributed as well as they may not in one plane. To have then the
rough estimation of cross section we depend on impact parameter, p.
The area πp2 is then the area within which a collision will result in scattering through an
angle θ. By differentiating it we can find dp corresponding to scattering between the
angle θ and θ + dθ. The corresponding solid angle is d = 2π sinθ dθ. The area
corresponding to dp is then
dA = 2πpdp = (π/2)s2sinθ dθ.
(15)
49
Angular momentum, Spin and Scattering
The ratio |dA/d)| is the probability for scattering into unit solid angle.
Let us see an example: Nuclear cross sections are expressed in barn (b). One barn is
equal to 10-28 m2. If Al27 is bombarded with slow neutrons, the most common reaction is
elastic scattering, or an (n,n) reaction. The cross section is about four times the
geometrical cross-section of the Al nucleus, about 2-3 b. But if we calculate geometrical
cross section for Al27it comes to about 0.7 b only. This result, which does not have any
explanation in classical physics, can well be expected on the basis of quantum mechanics.
The target is generally a thin foil in case of solids or a column of material in the case of
liquid or gases. The projectiles are atomic or subatomic particles. Hence the scattering
cross section does not exceed a few barns.
3.7 LABORATORY SYSTEM AND CENTER OF MASS SYSTEM
What is center of mass system (CM)?
It is the coordinate system in which the center of mass of the interacting particles is at
rest and remains at rest during and after interaction. It is an inertial system in which the
center of mass does not move.
The collision problem is solved most conveniently by transforming it to the center of
mass system. In this system, the result will be that the velocities have changed direction
by some angle.
Another important property is that in the CM system, the final momentum must be zero,
so the final velocities after collision are in the ratio of the masses.
What is laboratory coordinate system?
In the laboratory system the target is assumed to be at rest and the interacting particle is
in motion.
Note – it is easier to make measurements in laboratory system. Hence we generally make
measurements in laboratory system and do necessary calculations to express the result in
center of mass system.
50
Angular momentum
3.7.1 Connection between center of mass and laboratory system
Consider masses m1 and m2 are located at r1 and r2 with respect to a laboratory system.
We assume that the target is at rest in the laboratory frame. The location of center of mass
for two masses m1 and m2 located at r1 and r2 with respect to a laboratory system is
obtained from
rcm
m1r1 m2 r2
(m1 m2 )
(16)
Example-1. To find the mass of the composite particle when two particles are involved.
Let the particles undergoing motion have masses m1 and m2 respectively. Then the
Schrodinger equations for those particles are
[ħ2/2m] 21(r) = - [E1 – V1(r)] 1(r)
and
[ħ2/2m] 22(r) = - [E2 – V2(r)] 2(r)
and that for the composite system it is
[ħ2/2] 2 (r) = - [E – V (r)] (r)
(16)
where (r) =1(r) + 2(r), V(r) = V1(r) + V2(r) and E = E1 + E2
Comparison of the quantities on LHS suggests that
1
1
1
m1 m2
(17)
Example-2. Reduced mass of electron in hydrogen atom is calculated using m1 = 9.11 x
10-31 kg and m2 = 1.67 x 10-25 kg to give = 9.105 x 10 –31 kg. It suggests that if two
masses are widely different in magnitude, the reduced mass is almost equal to the mass of
the lighter particle. That is the reason the center of mass of the solar system lies within
the sun itself.
3.7.2 Relation between scattering angle in the two coordinate systems
To find the relation for the scattering angle, we move the laboratory system in the
direction of the incident particle with such a speed that the center of mass is relatively at
rest.
Thus the target which was stationary in the laboratory system now moves towards
projectile with velocity v2 such that
51
Angular momentum, Spin and Scattering
v2 = v – v1
v2 (m1 m2 )v1
and also
where v is velocity of projectile in the laboratory system and v1 is its velocity in the
center of mass system see Fig.4).
v1
m1
m1
v1
c
cm
v2
m2
m2
v2
Hence
Fig.4 Center of mass (CM) system
m1
v2 v
[m1 m2 ]
v2
Or
m2 v
[m1 m2 ]
If the scattering is elastic, the velocity remains the same after collision. Thus from the
figure 5, for the laboratory system
v2 cos c + v1 = v cos L
v2 sinc = v sinL
and
v2 sin c
v2 cos c v1
sin c
cos c
tan L
Or
Or
(18)
Where = v1/v2 = m1/m2
v
m1
m1
v
L
m2
52
Angular momentum
Fig.5 Laboratory coordinate system
One can find similarly relation for the cross section for the two systems as
L ( L )=
[1 2 2 cos c ]3/2
c (c )
[1 cos c ]
(19)
Check your progress-2
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) Find the location of center of mass of composite of particles m1, m2, m3
and m4 located at r1, r2, r3 and r4 from the origin of laboratory system.
-------------------------------------------------------------------------ii) Scattering cross section depends on a)………….b)……………and
inversely on c)………………..
3.8 SCATTERING BY A CONTROLLED POTENTIAL WAVE: ANALYSIS
AND PHASE SHIFT METHOD
3.8.1 Scattering by a square-well potential
Square well potential is defined as
V(r) = V0
= 0
when 0 < r < a
when r > a
where a is the width of the well.
If b is the impact parameter, n [
E V0 1/2
] and E = ½ mv02 then the scattering angle is
E
given by
= - 2 (i-r) = 2 [sin-1 (b/na) – sin-1(b/a)] for 0 <b< a
(20)
53
Angular momentum, Spin and Scattering
lim. ikz
eikr
(r )
e f ( )[ ]
r
r
The flux density is given by the second term. If is proportionality term depending on
angle then the scattered flux and incident wave flux are given respectively by
eikz
f ( )[ ]
r
2
eikr
f ( )[ ]
r
and
2
Let the detector be kept at a distance r and the area dA intercepts those waves, then the
flux
Or
eikr
f ( )[ ]
r
2
eikr
f ( )[ ]
r
2
dA = f ( )[eikr ]
dA
2
dA
r2
2
= f ( )[eikr ] d
(21)
Equation (21) represents the number of waves passing through the detector of area dA.
Here we have written
dA
= d, the solid angle subtended by the detector at the scatterer.
r2
Thus the differential scattering cross section is
2
( )d
f ( )[eikr ] d
f ( ) d
2
2
= [eikr ] d
(22)
3.8.2 Phase shift in scattering by a spherically symmetric potential
Let the Schrodinger equation for spherically symmetric potential V(r) be
2(r) + (2m/ħ2)[E – V(r)] (r) = 0
For spherical symmetry this equation has to be independent of
hence
where
Rl (r )
Pl (cos )
r
Rl ( r )
satisfies
r
54
Angular momentum
d 2 Rl [k 2 U (r ) l (l 1)]
Rl (r ) 0
dr 2
r2
with
(23)
k = (2mE/ ħ2) and U (r ) 2mV (r ) / ħ2
for r => we have
d 2 Rl
[k 2 ]Rl (r ) 0
dr 2
So
Pl cos Rl (r ) Pl cos
l
Or
l
Pl cos [
l
eikr
r
Cl eikr Dl eikr
]
r
r
If the incident wave is along z-axis we can write
eikz f ( )
eikr
r
For large distances from the scatterer
Or
eikz f ( )
C eikr Dl eikr
eikr
Pl (cos )[ l
]
r
r
r
l
eikz f ( )
eikr
l
Pl (cos )[bl sin( l )]
r
2
l
Where l is the phase shift of the lth partial wave. It is a measure by which
l
sin( l ) is displaced in relative to free particle function.
2
Thus the effect of spherical potential function is to shift the phase of the scattered wave.
The value of the quantity bl is found to be (i)l (2l 1)ei l this value helps in finding the
total scattering cross section.
(
4
) [2l 1]sin 2 l
k2 l
(24)
55
Angular momentum, Spin and Scattering
Note- Simple spherical potential fields are Columbian and gravitational. A microscopic
projectile when scattered by these must undergo a phase change. But it does not happen
with macro projectiles.
3.9 IMPACT PARAMETER
Impact parameter, b is the distance between the initial path of the projectile when it
started its journey towards the target and a parallel line that passes through the center of
the target particle. See Fig.6. is the scattering angle.
All those particles that lie at a distance between b and (b+db) are in an annular ring of
cross section 2bdb would be scattered such that
I 2 b db = - () I 2 sin d
( )
Or
b db
sin d
(25)
Negative sign indicates that decreases with increase in b.
m
v0
i
r1
b
r1
i
Fig.6
Classically, we can identify impact parameter p, as the distance of closest approach if the
particles would not interact, and had moved in straight line. This distance is not changed
by transformation to the CM system. In quantum mechanics, relative angular momentum
of the particles takes up the role of the impact parameter.
3.10 RELATION BETWEEN PHASE SHIFT AND LOGARITHMIC
DERIVATIVES
56
Angular momentum
This derivation is rather tricky. Let us start at r = 0 and proceed outward through the
region in which V(r) ≠ 0. The equation this situation can be written as
(1/r2)(d/dr)(r2dR/dr) + {2m[E - V(r)]/ħ2 - l(l + 1)/r2}R = 0.
Eventually, we will reach a point where V(r) can be considered zero and the potential
does no longer affect the solution. Let's call this distance r = a. Now we have, on the one
hand, our solution coming out from r = 0, and on the other, the general form of the radial
function when V = 0. The radial function can thus be written as
Rl(k.r) = Al[cos δl jl(k.r) - sin δl nl(k.r)].
Note that this is not the asymptotic solution at r = a, but includes the phase shifts in the
coefficients.
If u(r) is a wave function of the form of Rl, then the value of (1/u)(du/dr) will be
continuous at the boundary (a point where wave functions meet), since by definition of
wave functions both u and (du/dr) are continuous. The continuity of the logarithmic
derivative (1/u)(du/dr), should then give us the phase shifts. To understand it, let us
assume that γl is the value of the ratio (1/u)(du/dr) at r = a for the wave function for
values of r < a. Then, the continuity of the logarithmic derivative entails
k[cos δl
J l
n
(k.a) - sin δl l (k.a)]/[cos δl Jl(k.a) - sin δl nl(k.a)] = γl
r
r
(26)
Dividing numerator and denominator by cos δl and solving for tan δl, we can determine
the phase shifts in terms of the γl.
We get
tan δl = [k
J l
n
(k.a) - γlJl(k.a)]/[k l (k.a) - γlnl(k.a)]
r
r
(27)
From this equation we can find the cross section.
3.10.1 Effect of spin
57
Angular momentum, Spin and Scattering
When spin of the interacting particle is taken into account, each particle will have (2s+1)
spin Eigen functions.
Case I
When both the particles are in the same spin state with sz = m ħ, different states are given
by
U1(m)U2(m)
for s m -s.
Case II
Symmetric functions for the sum of products are
U1(m)U2(m) + U1(m)U2(m) for m m
Case III
There are s (2s+1) states with anti-symmetric functions consisting of differences of
products
U1(m)U2(m) - U1(m)U2(m) for m m
The total number of functions are (2s+1) + (s+1)(2s+1) + s(2s+1) = (2s+1)2. Of
these (s+1)(2s+1) are symmetric and s(2s+1) are anti-symmetric.
Example
1. Positive sign appears in differential cross section in (s+1)(2s+1)/(2s+1)2 =
(s+1/(2s+1) spin states. Thus for s = 1, positive sign appears in ratio 2/3; and for
s=2, it is 3/5 etc.
2. If s = 1/2 three states are symmetric and one is anti-symmetric.
Check your progress-3
Note:
a. Write your answers in the space provided below.
b. Compare your answers given at the end of the unit
i) What fraction of total functions will bear magnetic spin in differential
58
Angular momentum
cross section for two spin states?.
-------------------------------------------------------------------------ii) The incident particle may not suffer a scattering on traversing a
distance x is related to eNx. How would you claim it?
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3.11 BEHAVIOUR OF PHASE SHIFT AT LOW ENERGIES
3.11.1 Phase shifts at low energies
From the relation for we find that the phase shift completely determines the scattering.
The cross section is maximum when
l = /2, 3/2, 5/2 etc., and becomes zero at l =n.
When phase shifts are small the two functions fl and Fl of the following differential
equations
d 2 fl l (l 1)
[ 2 ] fl 0
d 2
d 2 Fl
V
l (l 1)
[1 ( )
]Fl 0
2
d
E
2
tend to be equal, i.e., fl = Fl.
So that
l
a
0
V
2
f l ( ) d 0
E
(28)
where a is the range of potential function. In such cases scattering becomes isotropic.
3.12 SCATTERING BY A HARD SPHERE
59
Angular momentum, Spin and Scattering
Hard sphere is defined by a potential
V (r) =
if r < a and V (r) = 0
if r > a
where 2a is the diameter of the sphere.
The Schrodinger equation for this sphere is
2(r) + (2m/ħ2)[E – V(r)] (r) = 0
We take the case r > a as the sphere is impermeable. Then this equation reduces to
2(r) + (2m/ħ2)[E] (r) = 0
Or
2(r) + k2 (r) = 0
where
k2 = (2m/ħ2)[E]
This equation written in spherical polar coordinates takes the form
1 d 2 dR
l (l 1)
[r
] [k 2
]R 0
2
r dr
dr
r2
(29)
To solve it, we substitute = kr.
So that
r = /k and
dr = d/k
So the equation takes the form
d
dR
l (l 1)
[ 2
] [ 2
]R 0
d
d
r2
Let us assume the probable solution to be R () = v ()
2
then
dv
dv
1
[ 2 l (l ) 2 ]v 0
2
d
d
2
(30)
This is standard Bessel equation of order (l+ ½ ). Its satisfactory solution is J(l+ ½ )()
R() =
J (l ½ ) ( )
J (l ½ ) ( )
2
(31)
where J (l ½ ) ( ) is called spherical Bessel function.
Remember that if J (l ) ( ) is a solution then J ( l ) ( ) has also to be a solution.
Asymptotic behaviour of Rl
60
Angular momentum
When , we can write
J ( l ½ ) ( )
Or
J l ( )
1
2
sin(
sin(
l
)
2
l
)
2
and corresponding
J l ( )
1
cos(
l
)
2
So that
Rl
A
C
sin(
l
B
l
) cos( )
2
2
[ J l ( ) cos l J l ( ) sin l ]
Or
Rl
Where
tan l = -(B/A)
(32)
Boundary conditionAt r = a the wave function vanishes i.e.,
J l ( ) cos l J l ( )sin l 0
Or
tan l
J l (ka )
J l (ka)
tan 0
J l /2 (ka )
tan(ka)
J l /2 (ka )
When l = 0,
Or
0 ka
and
4
sin 2 ka
2
k
(33)
For scattering involving extremely low energy ka <<1, so that
4 a 2 or a 2
(34)
61
Angular momentum, Spin and Scattering
This suggests that low energy scattering is mostly spherically symmetric.
When ka >> 1, we have
4
2
k
[2l 1]J l2 (ka)
l J 2 (ka) J 2 (ka)
l
l
2 a 2
or
(35)
The ratio of scattering cross sections for projectiles of high energy and low energy is
high/low 2a2/a2 = 2
with the above information you can try to show that ka >> 1 represents high-energy
situation and suggest why is r < a condition is forbidden in the case of solid sphere?
Born approximation
Whenever scattering effect is confined near the scattering centers, the amplitude of
scattered waves is very weak. For obtaining the scattering amplitudes in such cases
Born method is used.
Born assumed that the potential energy of interaction between projectile and the
scatterers is just a perturbation of first order. Born approximation thus can be
applied to those cases when colliding particle energy is very large in comparison to
energy of interaction. This is thus a supplement method to partial waves.
Ramsauer Scattering
It is an excellent example of particle scattering theory offered by the scattering of
low-energy electrons by atoms and molecules. He assumed atoms as hard spheres.
3.13 LET US SUM UP
Scattering is essentially a phenomenon resulting from an interaction of at least
one moving object with another stationary (or moving) object. After interaction it
is not only that energy and momentum of interacting particles undergo a change
but the phases also undergo a change.
62
Angular momentum
For macro objects a different phase is not noticeable and hence remained hidden
till the advent of quantum mechanics which discuss all aspects of dynamical
systems in detail. The studies can be undertaken essentially in two different ways
1) by considering motion in laboratory frame of reference and 2) investigating the
phenomenon in center of mass system. Since most of the observations are taken in
laboratory frame of reference an interrelation is studied between two frames.
Spin of the interacting particles modifies the formulation. Schrodinger proposed a
theory for scattering on the basis of wave packets.
Ultimately we studied simple phase relationships for spherically symmetric
potentials at low energies and just cite few other cases.
3.14 CHECK YOUR PROGRESS
1.
2.
i) The interference term is additional to individual amplitude components which is
2 Re f ( , ) f *( , ) , in the expression.
ii) since electrons are Fermions we retain negative interference term
( , ) [ f ( , )]2 [ f ( , )]2
m r m2 r2 m3r3 m4
i) rcm 1 1
. Simple principle being moment about center of
(m1 m2 m3 m4 )
mass must be zero.
ii) Scattering depends on a) sin2l, b) number of partial wave l and inversely on c)
energy of the projectile or k2. 2) Isotropic scattering means potential function
is spherically symmetric
3.
i) Substitute values of s in expression given in Case III we find that (s+1)(2s+1)
are symmetric functions which bear spin magnetism = 3x5 =15 functions
ii) I(x) = I0 eNx represents that I(x) out of I0 would suffer scattering for one
particle it is proportional to eNx
Reference books for further reading
1. Quantum mechanics: theory and applications, A Ghatak and S Loknathan, McMillan
63
Angular momentum, Spin and Scattering
2. Quantum mechanics, S L Gupta, V Kumar, H V Sharma and R C Sharma, Jai Prakash
Nath and Sons.
3. Quantum mechanics, L I Schiff, Mc Graw-Hill
4. Quantum mechanics, B K Agrawal and H Prasad, Prentice Hall of India
5. Introductory quantum mechanics and spectroscopy, K M Jain, South Asian Publishers
6. The Feynman Lecturers in Physics, Vol.III, R P Feynman, R B Leighten and M Sands,
Narosa Publishing House
7. Physics, D Halliday and R Resnick + Crane/Walker, John Wiley
8. Quantum Mechanics, V K Thankappan, Wiley Eastern Limited
9. Wikipedia, the free encyclopedia.
64
