notes21 - 5.73 Lecture#21 21 1 3D-Central Force Problems I...

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21 - 1 5.73 Lecture #21 revised October 21, 2002 @ 10:15 AM 1. define radial momentum 2. define orbital angular momentum 3. separate p 2 into radial and angular terms: 4. find Complete Set of Commuting Observables (CSCO) useful for block- diagonalizing H 5. 3D-Central Force Problems I All 2-Body, 3-D problems can be reduced to * a 2-D angular part that is exactly and universally soluble * a 1-D radial part that is system-specific and soluble by 1-D techniques in which you are now expert Next 3 lectures: Correspondence Principle Commutation Rules  → all matrix elements p r = r -1 q p - i h ( ) r L = r q × r p also L × L = i h L and L i , L j [ ] = i h k ε ijk L k p 2 = p r 2 + r −2 L 2 H , L 2 [ ] = H,L i [ ] = L 2 , L i [ ] = 0 H , L 2 , L i CSCO L,M L universal basis set separate radial part of H : H l (r) = p r 2 + V( r ) + h 2 l l +1 ( ) r 2 V l ( r ) effective radial potential 6. ALL Matrix Elements of Angular Momentum Components Derived from Commutation Rules. 7. Spherical Tensor Classification of all operators. 8. Wigner-Eckart Theorem all angular matrix elements of all operators. I hate differential operators. Replace them using exclusively simple Commutation Rule based Operator Algebra. general definition of angular momentum and of “vector operators” 1-D Schröd Eq. Read: C-TDL, pages 643-660 for next lecture. Roadmap
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21 - 2 5.73 Lecture #21 revised October 21, 2002 @ 10:15 AM Lots of derivations based on classical VECTOR ANALYSIS — much will be set aside as NONLECTURE Central Force Problems: 2 bodies where interaction force is along the vector r q 1 r q 2 1 2 CM r q 12 r q 1 r q 2 r q cm r r 1 r r 2 origin r q 2 = r q 1 + r q 12 r q 12 = r q 2 r q 1 = ˆ i x 2 x 1 ( ) + ˆ j y 2 y 1 ( ) + ˆ k z 2 z 1 ( ) r r q 12 = x 2 x 1 ( ) 2 + y 2 y 1 ( ) 2 + z 2 z 1 ( ) 2 [ ] 1 / 2 also C.M. Coordinate system r r 1 = r q 1 r q cm r 1 r = m 2 M [ ] v r 2 = r q 2 r q cm r 2 r = m 1 M [ ] H = H translation + H center of mass free translation of C of M of system of mass M = m 1 + m 2 motion of fictitious particle of mass in coordinate system with origin at C of M (CTDL page 713) µ = + m m m m 1 2 1 2 ( ) . . H P H P translation free rotation (no , dependence) constant = + ( ) + = µ + trans C M cm m m V V r 2 1 2 2 2 1 2 θ φ { LAB BODY free translation of system with respect to lab (not interesting) motion of particle of mass µ with respect to origin at c. of m.
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