Transcription of Harmonic Oscillator Physics - Reed College
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Harmonic Oscillator Physics - Reed College
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 . The existence of a minimum energy state ensured that no solutions could have negative energy and was used to define 0 1 : 1.
p 2 2 r m! ~ x 2 e m!x 2 2~ = A 2 p 2 4 m! ~ x2 2 e m!x 2 2~: (9.21) The pattern continues { we always have some polynomial in xmultiplying the exponential factor. That polynomial, for the nthwave function is called H n, the nthHermite polynomial. In the dimensionless variable ˘= p m! ~ x, we can read o the rst two { H 1(˘) = 2˘, and H 2 ...
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