Orbital Angular Momentum

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 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 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” P460 - angular momentum

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 P460 - angular momentum

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 P460 - angular momentum

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 P460 - angular momentum

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 P460 - angular momentum

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 P460 - angular momentum

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 P460 - angular momentum

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

16. 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 momentum

17. 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

18. 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

19. 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) P460 - angular momentum

20. 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

21. 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

22. 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