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# notes12 - 5.73 Lecture#12 Matrix Solution of Harmonic...

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12 - 1 5.73 Lecture #12 updated 9/20/02 11:44 AM 1. Assumptions 2. x nm and p nm in terms of (E n –E m ) 3. x nm in terms of p nm 4. Block Diagonalize x, p, H 5. Lowest quantum number must exist (call it 0) explicit values for 6. Recursion relationship for x nn±1 and p nn±1 7. Magnitudes and phases for x nn±1 and p nn±1 8. Possibility of noncommunicating blocks along diagonal of H, x, p eliminated Matrix Solution of Harmonic Oscillator Last time: * * * * T A T T T xT T φ φ = = ( ) a a i i h N 1 0 0 0 0 0 0 O M eigenbasis T T t column of matrix of function of matrix is given by f Discrete Variable Representation: Matrix representation for 1- D problem 1 Ni i T any TODAY: Derive all matrix elements of x , p , H from [ x,p ] commutation rule and definition of H . Example of how one can get matrix results entirely from commutation rule definitions (e.g. of an angular momentum: J 2 , J x , J y , J z and Wigner-Eckart Theorem) NO WAVEFUNCTIONS, NO INTEGRALS, ALL MAGIC! Outline of steps: * * ˆ ˆ * ˆ ˆ H p x H i = + [ ] = 2 2 2 2 m k x eigen basis exists for * x,p and p are Hermitian (real expectation values) h See CTDL pages 488-500 for similar treatment. You will never use this methodology - only the results! IN MORE ELEGANT NOTATION x 01 2 and p 01 2

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12 - 2 5.73 Lecture #12 updated 9/20/02 11:44 AM 1. recall assumptions 2. x and p matrix elements derived from Comm. Rules x , H [ ] = x , p 2 2 m + 1 2 k x 2 = 1 2 m x , p 2 [ ] = 1 2 m p x , p [ ] + x , p [ ] p ( ) x , p [ ] = i h ** x ,H [ ] = p 2 m 2 i h = i h m p p = m i h x , H [ ] p nm = m i h x n l H l m H n l x l m ( ) l x nm = i k h p n l H l m H n l p l m ( ) l but we know that some basis set must exist where H is diagonal. Use it implicitly: replace H m by E m δ m p nm = m i h x nm E m E n x nm ( ) p nm = m i h x nm E m E n ( ) p nn = 0 (but, in addition, if H has a degenerate eigenvalue, then p nm = 0 if E n = E m ) similarly for x nm = i h k p
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notes12 - 5.73 Lecture#12 Matrix Solution of Harmonic...

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