P460 - angular momentum1 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 momentum2 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 

P460 - angular momentum3 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 when solving angular part of S.E. and so and know eigenvalues for L 2 and L z with spherical harmonics being eigenfunctions

P460 - angular momentum4 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 but there is another operator that can be simultaneously diagonalized

P460 - angular momentum5 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 momentum6 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 -1 A=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”

P460 - angular momentum7 Simple example Discrete group. Properties of group (its “arithmetic”) contained in Table If represent each term by a number, and group combination is normal multiplication or can represent by matrices and use normal matrix multiplication

P460 - angular momentum8 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

P460 - angular momentum9 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

P460 - angular momentum10 For the Rotation group O(3) by inspection as: one gets a representation for angular momentum satisfies Group Algebra Another group SU(2) also satisfies same Algebra. 2x2 Unitary transformations (matrices) with det=1 (gives S=special). SU(n) has n 2 -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 momentum11 Eigenvalues “Group Theory” Use the group algebra to determine the eigenvalues for the two diagonalized operators L z and L 2 ( Already know the answer) Have constraints from “geometry”. eigenvalues of L 2 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 Define raising and lowering operators (ignore Plank’s constant for now). “Raise” m-eigenvalue while keeping l-eiganvalue fixed

P460 - angular momentum12 Eigenvalues “Group Theory” operates on a 1x2 “vector” (varying m) raising or lowering it

P460 - angular momentum13 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

P460 - angular momentum14 Eigenvalues Done in different ways (Gasior,Griffiths,Schiff) Start with two diagonalized operators L z and L 2. where m and are not yet known Define raising and lowering operators (in m) and easy to work out some relations Assume if g is eigenfunction of L z and L 2.,L + g is also an eigenfunction new eigenvalues (and see raises and lowers value)

P460 - angular momentum15 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 L z go from -l to l in integer steps (N steps)

P460 - angular momentum16 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

P460 - angular momentum17 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)

P460 - angular momentum18 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