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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 - 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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