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# Lect20_SHO2 - The Story So Far Start from the Hamiltonian...

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Lecture 20: QM Harmonic Oscillator II PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore The Story So Far Start from the Hamiltonian: Introduce dimensionless Units: Dimensionless Hamiltonian: Introduce new variables: Hamiltonian becomes: 2 2 2 2 1 2 X m m P H ! + = ! X X = P P h ! = ! h H H = ! " m h = 2 2 2 1 2 1 X P H + = Harmonic Oscillator Length ( ) P i X A + = 2 1 ( ) P i X A ! = 2 1 A , A [ ] = 1 2 1 + = A A H

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Eigenvalue Spectrum Define the eigenstates: Lowering and Raising operators: Energy lower bound: Coefficients: ! ! ! = H ! ! " ! ! # = # , A , H [ ] = A H A = A H " 1 ( ) H A = A H + 1 ( ) H A " ( ) = " # 1 ( ) A " ( ) ( ) ! ! ! 1 + = A A H 1 ! = " # # # c A 1 + = ! " " " d A ! ! ! H = " = 1 2 + " A ! A " = 1 2 + A " 2 2 1 ! " # 2 1 2 ! = " " c 2 1 2 + = ! ! d The Big Trick Start from Logic: if | ! " exists, then either c ! vanishes or | ! -1 " exists c ! only vanishes for ! =1/2 Choosing the ground state as ! =1/2 is therefore self-consistent Any other choice is not self-consistent, as a lower state would have to exist If | ! =1/2 " exists then the states with ! =1/2, 3/2 , 5/2, 7/2,… exist Ladder of equally spaced states If any other state exists, then a state below ! =1/2 must exist. Since the latter is impossible non-
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Lect20_SHO2 - The Story So Far Start from the Hamiltonian...

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