385lec8 - PHYS 385 Lecture 8 - Schrdinger equation with...

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PHYS 385 Lecture 8 - Schrödinger equation with interactions 8 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 8 - Schrödinger equation with interactions What's important : energy operator SE with potential energy Schrödinger equation in three dimensions Text : Gasiorowicz, Chaps. 3 Energy operator Recall again the form of the Schrödinger equation for a free particle in one dimension: i h ( x , t ) t = - h 2 2 m 2 ( x , t ) x 2 (1) In the previous lecture, we established that the momentum operator p op could be represented by p = - i h x (2) when operating on wavefunctions in position-space, ( x , t ). From now on, we drop the "op" subscript unless we need it for clarity. Squaring this (not complex square!), p 2 = - h 2 2 x 2 (3) which bears a resemblance to the right-hand side of Eq. (1). Substituting, (1) becomes i h ( x , t ) t = p 2 2 m ( x , t ) Now, p 2 / 2 m is just the non-relativistic kinetic energy, so the action of the right hand side is to operate on ( x , t ) with the kinetic energy operator. Note that we say "operate
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385lec8 - PHYS 385 Lecture 8 - Schrdinger equation with...

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