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385lec20

# 385lec20 - PHYS 385 Lecture 20 Schrdinger equation in 3D...

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PHYS 385 Lecture 20 - Schrödinger equation in 3D 20 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 20 - Schrödinger equation in 3D What's important : Schrödinger equation in 3D angular momentum operators Text : Gasiorowicz, Chap. 10 3D Schrödinger equation As discussed in Lec. 17, the time-independent Schrödinger equation for a free particle in three dimensions reads: - h 2 2 m d 2 dx 2 + d 2 dy 2 + d 2 dz 2 u E ( x , y , z ) = E u E ( x , y , z ). (1) Adding a time-independent potential can easily be accommodated through - h 2 2 m d 2 dx 2 + d 2 dy 2 + d 2 dz 2 u E ( x , y , z ) + V ( x , y , z ) u E ( x , y , z ) = E u E ( x , y , z ) (2) Now, for many situations of interest, the potential: is between two objects, both of whose motion must be considered depends only on the radial separation r , so-called central potentials. Two-particle wavefunctions in one-dimension were introduced in Lec. 16, where we showed that if the potential energy depended only on the relative separation x rel , the wavefunction could be written as a product of independent wavefunction for the relative motion and the cm motion. The same is true for three dimensions for central potentials V ( r ). To review the notation ( bold indicates vectors): R cm = ( m 1 r 1 + m 2 r 2 ) / ( m 1 + m 2 ) and (3) r = r 1 - r 2 . With this replacement, the arguments of the plane wave p 1 r 1 + p 2 r 2 = P cm R cm + pr , (4) where the total and relative wavevectors are P cm = p 1 + p 2 and p = ( m 2 p 1 - m 1 p 2 ) / ( m 1 + m 2 ). (5) The total kinetic energy then becomes E = P cm 2 / 2 M total + p 2 / 2 μ . (6)

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PHYS 385 Lecture 20 - Schrödinger equation in 3D 20 - 2 ©2003 by David Boal, Simon Fraser University.
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385lec20 - PHYS 385 Lecture 20 Schrdinger equation in 3D...

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