Lect29_CentralPot - Lecture 29: Motion in a Central...

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Lecture 29: Motion in a Central Potential Phy851 Fall 2009
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Side Remarks Counting quantum numbers: – 3N quantum numbers to specify a basis state for N particles in 3-dimensions – It will go up to 5N when we include spin – When does it work: • All of the standard basis choices – Position eigenstates, Momentum eigenstates, angular momentum eigenstates, … • Any basis formed from energy eigenstates of an analytically solvable system: – Harmonic oscillator states, hydrogen orbitals, ... – These problems are solvable due to a high degree of symmetry • Any basis formed from eigenstates of an exactly solvable system plus a weak symmetry breaking perturbation – We can watch the levels evolve as we increase the perturbation strength, and therefore keep track of the quantum numbers – When it does not work • Strongly interacting systems with minimal symmetry – These are problems that you could only solve numerically, they won’t be encountered in class or in textbooks
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`And versus or paradigm Hilbert subspace Hierarchy: – To specify the state of the full system, we must specify a state in AND a state in – To specify the state of the relative motion we may specify a state entirely in OR a state entirely in OR a state partially in both ) ( ) ( R C I I I = ) ( ) ( ) ( R continuum R bound R I I I + = ) ( ) ( , R C H H ) ( ) ( , R continuum R bound H H H ) ( R H ) ( C H ) ( R bound H ) ( R continuum H
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The center-of-mass motion is that of a free particle We thus only need to determine to state of relative motion: As long as V depends only r and not on θ , φ then simultaneous eigenstates of H r , L 2 and L z exist: Review of Separation of Variables and Angular Momentum ) ( 2 2 2 2 2 R V R L P H r r + + = μ r CM H H
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This note was uploaded on 12/02/2010 for the course PHYSICS 851 taught by Professor M.moore during the Fall '09 term at Michigan State University.

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Lect29_CentralPot - Lecture 29: Motion in a Central...

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