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## Orbital Angular Momentum

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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))

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Orbital Angular Momentum

z

f

- 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

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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

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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

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Commutation Relationships

- but there is another operator that can be simultaneously diagonalized (Casimir operator)

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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)

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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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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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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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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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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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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)

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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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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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Eigenvalues “Group Theory”

- operates on a 1x2 “vector” (varying m) raising or lowering it

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Can also look at matrix representation for 3x3 orthogonal (real) matrices

- Choose Z component to be diagonal gives choice of matrices
- can write down (need sqrt(2) to normalize)
- and then work out X and Y components

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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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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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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
- repeat for the lowest state
- eigenvalues of Lz go from -l to l in integer steps (N steps)

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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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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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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

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