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HarmonicOscillator - Harmonic Oscillator From a and a to...

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Harmonic Oscillator: From a and a to the position representation Michele Kotiuga Physics 137A Spring 2010 In lecture the harmonic oscillator potential was treated using creation and annihilation operators: a and a . In Section 4.7 of Bransden and Joachain the harmonic oscillator potential is solved in the position representation - i.e. we found the wavefunction as a function of position: ψ ( x ). The method is quite messy and the solution is found from recursion relations: ψ n ( x ) = N n e - α 2 x 2 / 2 H n ( αx ) (1) Where N n is a constant ensuring that the wavefunction is normalized and H n is a Hermite polynomial. The Hermite polynomials can be generated from the formula: H n ( ξ ) = e ξ 2 / 2 ξ - d n e - ξ 2 / 2 (2) On the other hand, using creation and annihilation operators, the wavefuction takes the form: ψ n = ( a ) n n ! | 0 > (3) Where | 0 > is the ground state.I will show that these two wavefucntions are equivalent, by changing to the position representation. Writing a and a in terms of x and p : a = r h x + ip
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