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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 - PowerPoint PPT Presentation

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P460 - angular momentum*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 equations separation constanteigenvalues and eigenfunctions for this can be found by solving the differential equation using series solutionsbut also can be solved algebraically. This starts by assuming L is conserved (true if V(r))

P460 - angular momentum

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

zf

P460 - angular momentum

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

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

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

P460 - angular momentum

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 algebraicallysimilar to what was done for harmonic oscillatoran example of a group theory application. Also shows how angular momentum terms are combinedthe 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 doesnt have any particles with that groups properties)

P460 - angular momentum

P460 - angular momentum*Sidenote:Group Theory A very simplified introductionA set of objects form a group if a combining process can be defined such that1. If A,B are group members so is AB2. The group contains the identity AI=IA=A3. There is an inverse in the group A-1A=I4. Group is associative (AB)C=A(BC)group not necessarily commutative Abelian non-AbelianCan often represent a group in many ways. A table, a matrix, a definition of multiplication. They are then isomorphic or homomorphic

P460 - angular momentum

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

P460 - angular momentum*Continuous (Lie) Group:Rotations Consider the rotation of a vector

R is an orthogonal matrix (length of vector doesnt change). All 3x3 real orthogonal matrices form a group O(3). Has 3 parameters (i.e. Euler angles)O(3) is non-Abelianassume angle change is small

P460 - angular momentum

P460 - angular momentum*Rotations Also need a Unitary Transformation (doesnt 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

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

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

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 cant be greater than the total (and since z is arbitrary, reverse also true)

The X and Y components arent 0 (except if L=0) but cant be diagonalized and so ~indeterminate with a range of possible values

P460 - angular momentum

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

P460 - angular momentum

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

P460 - angular momentum

P460 - angular momentum* Can also look at matrix representation for 3x3 orthogonal (real) matricesChoose Z component to be diagonal gives choice of matrices

P460 - angular momentum

P460 - angular momentum* Can also look at matrix representation for 3x3 orthogonal (real) matricescan write down L+- (need sqrt(2) to normalize) and then work out X and Y components

P460 - angular momentum

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

P460 - angular momentum

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

P460 - angular momentum

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 knownDefine raising and lowering operators (in m) and easy to work out some relations

P460 - angular momentum

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

P460 - angular momentum*Eigenvalues There must be a highest and lowest value as cant have the z-component be greater than the total

For highest state, let l be the maximum eigenvalue

can easily show

P460 - angular momentum

P460 - angular momentum*Eigenvalues There must be a highest and lowest value as cant 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

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=-mvery useful when adding together angular momentums and building up eigenfunctions. Gives Clebsch-Gordon coefficients

P460 - angular momentum

P460 - angular momentum*Eigenfunctions in spherical coordinates if l=integer can determine eigenfunctionsknowing 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

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 equationnote power of l: l=2 will have

P460 - angular momentum

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P460 - 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)) 2 2 2 mr L 0 ] , [ 0 L H dt L d

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