Introduction to Quantum Theory of Angular Momentum

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Introduction to Quantum Theory of Angular Momentum
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Angular Momentum

• AM begins to permeate QM when you move from 1-d
  to 3-d
• This discussion is based on postulating rules for
  the components of AM
• Discussion is independent of whether spin,
  orbital angular momenta, or total momentum.

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Definition

• An angular momentum, J, is a linear operator with
  3 components (Jx, Jy, Jz) whose commutation
  properties are defined as

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Or in component form
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Convention

• Jz is diagonal
• For example

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Therefore
Where jmgt is an eigenket h-bar m is an
eigenvalue For a electron with spin up
Or spin down
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Definition
These Simple Definitions have some major
consequences!
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THM
Proof
QED
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Raising and Lowering Operators
Lowering Operator
Raising Operator
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Product of J and J
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Fallout
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Proof that J is the lowering operator
It is a lowering operator since it works on a
state with an eigenvalue, m, and produces a new
state with eigenvalue of m-1
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J2,Jz0 indicates J2 and Jz are simultaneous
observables
Since Jx and Jy are Hermitian, they must have
real eigenvalues so l-m2 must be positive! l is
both an upper and LOWER limit to m!
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Let msmalllower bound on m andlet mlargeupper
bound on m
mlarge cannot any larger
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Final Relation
So the eigenvalue is mlarge(mlarge 1) for any
value of m
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Four Properties
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Conclusions

• As a result of property 2), m is called the
  projection of j on the z-axis
• m is called the magnetic quantum number because
  of the its importance in the study of atoms in a
  magnetic field
• Result 4) applies equally integer or half-integer
  values of spin, or orbital angular momentum

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END OF LECTURE 1
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Matrix Elements of J
Indicates a diagonal matrix
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Theorems
And we can make matrices of the eigenvalues, but
these matrices are NOT diagonal
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Fun with the Raising and Lowering Operators
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A matrix approach to Eigenvalues
If j0, then all elements are zero! B-O-R-I-N-G!
Initial m
j 1/2
final m
What does J look like?
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Using our relations,
Answer
Pauli Spin Matrices
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J1, An Exercise for the Students
Hint
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Rotation Matices

• We want to show how to rotate eigenstates of
  angular momentum
• First, lets look at translation
• For a plane wave

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A translation by a distance, A, then looks like
translation operator
Rotations about a given axis commute, so a finite
rotation is a sequence of infinitesimal
rotations Now we need to define an operator for
rotation that rotates by amount, q, in direction
of q
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So
Where n-hat points along the axis of rotation
Suppose we rotated through an angle f about the
z-axis
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Using a Taylor (actually Maclaurin) series
expansion
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What if f 2p?
The naïve expectation is that thru 2p and no
change. This is true only if j integer. This
is called symmetric BUT for ½ integer, this is
not true and is called anti-symmetric
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Let j1/2 (for convenience it could be any value
of j)
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Using the sine and cosine relation
And it should be no surprise, that a rotation of
b around the y-axis is
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Consequences

• If one rotates around y-axis, all real numbers
• Whenever possible, try to rotate around z-axis
  since operator is a scalar
• If not possible, try to arrange all non-diagonal
  efforts on the y-axis
• Matrix elements of a rotation about the y-axis
  are referred to by

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And
Wigners Formula (without proof)
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Certain symmetry properties of d functions are
useful in reducing labor and calculating rotation
matrix
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Coupling of Angular Momenta

• We wish to couple J1 and J2

• From Physics 320 and 321, we know

• But since Jz is diagonal, m3m1m2

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Coupling contd

• The resulting eigenstate is called
• And is assumed to be capable of expansion of
  series of terms each of with is the product of 2
  angular momentum eigenstates conceived of riding
  in 2 different vector spaces
• Such products are called direct products

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Coupling contd

• The separateness of spaces is most apparent when
  1 term is orbital angular momentum and the other
  is spin
• Because of the separateness of spaces, the direct
  product is commutative
• The product is sometimes written as

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Proof of commutative property
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The expansion is written as
Is called the Clebsch-Gordan coefficient Or
Wigner coefficient Or vector coupling coefficient
Some make the C-G coefficient look like an inner
product, thus
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A simple formula for C-G coefficients

• Proceeds over all integer values of k
• Begin sum with k0 or (j1-j2-m3) (which ever is
  larger)
• Ends with k(j3-j1-j2) or kj3m3 (which ever is
  smaller)
• Always use Stirlings formula log (n!) nlog(n)

Best approach use a table!!!
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• What if I dont have a table?
• And Im afraid of the simple formula?
• Well, there is another path a 9-step path!

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9 Steps to Success

1. Get your values of j1 and j2
2. Identify possible values of j3
3. Begin with the stretched cases where j1j2j3
   and m1j1, m2j2 , and m3j3, thus j3 m3gtj1
   m1gtj2 m2gt
4. From J3J1J2,, it follows that the lowering
   operator can be written as J3J1J2

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9 Steps to Success, contd

• Operate J3j3 m3gt(J1J2 )j1 m1gtj2 m2gt
• Use
• Continue to lower until j3j1-j2, where m1-j1
  , m2 -j2, and m3 -j3
• Construct j3 m3 gt j1j2 -1 j1j2-1gt so that
  it is orthogonal to j1j2 j1j2-1gt
• Adopt convention of Condon and Shortley,
• if j1 gt j2 and m1 gt m2 then
• Cm1 m2j1 j2 j3 gt 0
• (or if m1 j1 then coefficient positive!)

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9 Steps to Success, contd

• Continue lowering and orthogonalizin until
  complete!
• Now isnt that easier?
• And much simpler
• You dont believe me Im hurt.
• I know! How about an example?

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A CG Example j1 1/2 and j2 1/2
Step 1
Step 2
Step 3
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Steps 4 and 5 and 6-gt
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Step 7Keep lowering
As low as we go
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An aside to simplify notation
Now we have derived 3 symmetric states
Note these are also symmetric from the standpoint
that we can permute space 1 and space 2 Which is
1? Which is 2? I am not a number I am a free
man!
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The infamous step 8

• Construct j3 m3 gt j1j2 -1 j1j2-1gt so that
  it is orthogonal to j1j2 j1j2-1gt
• j1j21 and j1j2-10 for this case so we want to
  construct a vector orthogonal to 1 0gt
• The new vector will be 0 0gt

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Performing Step 8
An orthogonal vector to this could be
or
Must obey Condon and Shortley if m1j1,, then
positive value j11/2 and gt represents m ½ ,
so only choice is
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Step 9 The End
This state is anti-symmetric and is called the
singlet state. If we permute space 1 and space
2, we get a wave function that is the negative of
the original state.
These three symmetric states are called the
triplet states. They are symmetric to any
permutation of the spaces
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A CG Table look up Problem

• Part 1
• Two particles of spin 1 are at rest in a
  configuration where the total spin is 1 and the
  m-component is 0. If you measure the z-component
  of the second particle, what values of might you
  get and what is the probability of each
  z-component?

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CG Helper Diagram
j3 m3 C
m1 m2
It is understood that a C means square root of
C (i.e. all radicals omitted)
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Solution to Part 1

• Look at 1 x 1 table
• Find j3 1 and m3 0
• There 3 values under these

m1 m2
1 -1 1/2
0 0 0
-1 1 -1/2
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So the final part
m2 C Prob
-1 1/2 ½
0 0 0
1 -1/2 ½
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Part 2

• An electron is spin up in a state, y5 2 1 , where
  5 is the principle quantum number, 2 is orbital
  angular momentum, and 1 is the z-component.
• If you could measure the angular momentum of the
  electron alone, what values of j could you get
  and their probabilities?

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Solution

• Look at the 2 x ½ table since electron is spin ½
  and orbital angular momentum is 2
• Now find the values for m11 and m21/2
• There are two values across from these
• 4/5 which has j3 5/2
• -1/5 which has j3 3/2
• So j35/2 has probability of 4/5
• So j3 3/2 has probability of 1/5