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.

+ m!xg(x) dx = Z 1 1 ~ @f(x) @x g(x) + f(x) m!xg(x) dx = Z 1 1 (a f(x)) g(x)dx; (9.5) (integration by parts) or, in words, we can \act on g(x) with a or act on f(x) with a" in the inner product. 9.1 Normalization The state n that comes from napplications of an + is not normalized, nor does the eigenvalue form of the time-independent Schr ...

Loading..

Tags:

  Physics, Oscillators, Harmonics, Harmonic oscillator physics

Information

Related search queries