1. Orbital Angular Momentum
In classical mechanics, conservation of angular momentum L is sometimes treated by an effective (repulsive) potential
Soon we will solve the 3D Schr. Eqn. The R equation will have an angular momentum term which arises from the Theta equation’s separation constant
eigenvalues and eigenfunctions for this can be found by solving the differential equation using series solutions
but also can be solved algebraically. This starts by assuming L is conserved (true if V(r))
P460 - angular momentum

2. Orbital Angular Momentum
Look at the quantum mechanical angular momentum operator (classically this “causes” a rotation about a given axis)
look at 3 components
operators do not necessarily commute z f
P460 - angular momentum

3. Side note Polar Coordinates
Write down angular momentum components in polar coordinates (Supp 7-B on web,E&R App M)
and with some trig manipulations
but same equations will be seen when solving angular part of S.E. and so
and know eigenvalues for L2 and Lz with spherical harmonics being eigenfunctions
P460 - angular momentum

4. Commutation Relationships
Look at all commutation relationships
since they do not commute only one component of L can be an eigenfunction (be diagonalized) at any given time
P460 - angular momentum

5. Commutation Relationships
but there is another operator that can be simultaneously diagonalized (Casimir operator)
P460 - angular momentum

6. Group Algebra
The commutation relations, and the recognition that there are two operators that can both be diagonalized, allows the eigenvalues of angular momentum to be determined algebraically
similar to what was done for harmonic oscillator
an example of a group theory application. Also shows how angular momentum terms are combined
the group theory results have applications beyond orbital angular momentum. Also apply to particle spin (which can have 1/2 integer values)
Concepts later applied to particle theory: SU(2), SU(3), U(1), SO(10), susy, strings…..(usually continuous)…..and to solid state physics (often discrete)
Sometimes group properties point to new physics (SU(2)-spin, SU(3)-gluons). But sometimes not (nature doesn’t have any particles with that group’s properties)
P460 - angular momentum

7. Sidenote:Group Theory
A very simplified introduction
A set of objects form a group if a “combining” process can be defined such that
1. If A,B are group members so is AB
2. The group contains the identity AI=IA=A
3. There is an inverse in the group A-1A=I
4. Group is associative (AB)C=A(BC)
group not necessarily commutative
Abelian
non-Abelian
Can often represent a group in many ways. A table, a matrix, a definition of multiplication. They are then “isomorphic” or “homomorphic”
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8. Simple example
Discrete group. Properties of group (its “arithmetic”) contained in Table
Can represent each term by a number, and group combination is normal multiplication
or can represent by matrices and use normal matrix multiplication
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9. Continuous (Lie) Group:Rotations
Consider the rotation of a vector
R is an orthogonal matrix (length of vector doesn’t change). All 3x3 real orthogonal matrices form a group O(3). Has 3 parameters (i.e. Euler angles)
O(3) is non-Abelian
assume angle change is small
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10. Rotations
Also need a Unitary Transformation (doesn’t change “length”) for how a function is changed to a new function by the rotation
U is the unitary operator. Do a Taylor expansion
the angular momentum operator is the “generator” of the infinitesimal rotation
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11. For the Rotation group O(3) by inspection as:
one gets a representation for angular momentum (notice none is diagonal; will diagonalize later)
satisfies Group Algebra
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12. Group Algebra
Another group SU(2) also satisfies same Algebra. 2x2 Unitary transformations (matrices) with det=1 (gives S=special). SU(n) has n2-1 parameters and so 3 parameters
Usually use Pauli spin matrices to represent. Note O(3) gives integer solutions, SU(2) half-integer (and integer)
P460 - angular momentum

13. Eigenvalues “Group Theory”
Use the group algebra to determine the eigenvalues for the two diagonalized operators Lz and L2 Already know the answer
Have constraints from “geometry”. eigenvalues of L2 are positive-definite. the “length” of the z-component can’t be greater than the total (and since z is arbitrary, reverse also true)
The X and Y components aren’t 0 (except if L=0) but can’t be diagonalized and so ~indeterminate with a range of possible values
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14. Eigenvalues “Group Theory”
Define raising and lowering operators (ignore Plank’s constant for now). “Raise” m-eigenvalue (Lz eigenvalue) while keeping l-eiganvalue fixed
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15. Eigenvalues “Group Theory”
operates on a 1x2 “vector” (varying m) raising or lowering it
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16. Choose Z component to be diagonal gives choice of matrices
Can also look at matrix representation for 3x3 orthogonal (real) matrices
Choose Z component to be diagonal gives choice of matrices
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17. Can also look at matrix representation for 3x3 orthogonal (real) matrices
can write down L+- (need sqrt(2) to normalize) and then work out X and Y components
P460 - angular momentum

18. Can also look at matrix representation for 3x3 orthogonal (real) matrices. Work out X and Y components
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19. Can also look at matrix representation for 3x3 orthogonal (real) matrices. Work out L2
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20. Eigenvalues Done in different ways (Gasior,Griffiths,Schiff)
Start with two diagonalized operators Lz and L2.
where m and l are not yet known
Define raising and lowering operators (in m) and easy to work out some relations
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21. Eigenvalues
Assume if g is eigenfunction of Lz and L2. ,L+g is also an eigenfunction
new eigenvalues (and see raises and lowers value)
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22. Eigenvalues
There must be a highest and lowest value as can’t have the z-component be greater than the total
For highest state, let l be the maximum eigenvalue
can easily show
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23. Eigenvalues
There must be a highest and lowest value as can’t have the z-component be greater than the total
repeat for the lowest state
eigenvalues of Lz go from -l to l in integer steps (N steps)
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24. Raising and Lowering Operators
can also (see Gasior,Schiff) determine eigenvalues by looking at
and show
note values when l=m and l=-m
very useful when adding together angular momentums and building up eigenfunctions. Gives Clebsch-Gordon coefficients
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25. Eigenfunctions in spherical coordinates
if l=integer can determine eigenfunctions
knowing the forms of the operators in spherical coordinates
solve first
and insert this into the second for the highest m state (m=l)
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26. Eigenfunctions in spherical coordinates
solving
gives
then get other values of m (members of the multiplet) by using the lowering operator
will obtain Y eigenfunctions (spherical harmonics) also by solving the associated Legendre equation
note power of l: l=2 will have
P460 - angular momentum