Lect28_TwoBodyProb - Lecture 28 The Quantum Two-body...

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Lecture 28: The Quantum Two-body Problem Phy851 Fall 2009
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Two interacting particles Consider a system of two particles with no external fields By symmetry, the interaction energy can only depend on the separation distance: From our experience with Classical Mechanics, we might want to treat separately the Center-of-mass and relative motion: Center-of-mass coordinate: Relative coordinate: This is recommended because the potential depends only on the relative coordinate: H = P 1 2 2 m 1 + P 2 2 2 m 2 + V r R 1 r R 2 ( ) 2 1 2 2 1 1 m m R m R m R CM + + = r r r 2 1 R R R r r r = V r R 1 r R 2 ( ) = V ( R )
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Center-of-mass and relative momentum How do we go about finding the center-of- mass and relative-motion momentum operators: Can we use: Answer: No, this is very wrong! Lets try instead to use what we know from classical mechanics: 2 1 P P P r r r = 2 2 1 1 V m V m r r + = 2 1 2 2 1 1 m m P m P m P CM + + = r r r CM CM V M P r r = 2 1 m m M + = 2 1 2 1 m m m m + = μ 2 1 2 2 1 1 m m V m V m R dt d V CM CM + + = = r r r r 2 1 P P P CM r r r + = 2 1 V V R dt d V r r r r = = ?
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