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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online