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

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

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

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

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

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

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” P460 - angular momentum

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 P460 - angular momentum

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 momentum

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 momentum

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 P460 - angular momentum

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

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 P460 - angular momentum

Eigenvalues “Group Theory” Define raising and lowering operators (ignore Plank’s constant for now). “Raise” m-eigenvalue (Lz eigenvalue) while keeping l-eiganvalue fixed P460 - angular momentum

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

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 P460 - angular momentum

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

Can also look at matrix representation for 3x3 orthogonal (real) matrices. Work out X and Y components P460 - angular momentum

Can also look at matrix representation for 3x3 orthogonal (real) matrices. Work out L2 P460 - angular momentum

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 P460 - angular momentum

Eigenvalues Assume if g is eigenfunction of Lz and L2. ,L+g is also an eigenfunction new eigenvalues (and see raises and lowers value) P460 - angular momentum

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 P460 - angular momentum

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) P460 - angular momentum

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 momentum

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 momentum

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