Transcription of Harmonic Oscillator Physics
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Harmonic Oscillator Physics
Physics 342 lecture 9. Harmonic Oscillator Physics lecture 9. Physics 342. quantum mechanics I. Friday, February 12th, 2010. For the Harmonic Oscillator potential in the time-independent Schro dinger equation: 1 2 d (x). 2.. ~ + m x (x) = E (x), 2 2 2. ( ). 2m dx2. we found a ground state m x2. 0 (x) = A e 2~ ( ). with energy E0 = 1. 2 ~ . Using the raising and lowering operators 1. a+ = ( i p + m x). 2~m . ( ). 1. a = (i p + m x), 2~m . we found we could construct additional solutions with increasing energy using a+ , and we could take a state at a particular energy E and construct solutions with lower energy using a.
Lecture 9 Physics 342 Quantum Mechanics I Friday, February 12th, 2010 ... 9.3.2 Quantum Case Referring to the de nition of the a + and a operators in terms of xand p, we can invert and nd xand pin terms of a + and a { these are all still operators, but we are treating them algebraically. The inversion is simple x= r ~ 2m! (a + + a) p= i r ~m! 2 ...
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