Chapter 6 Lecture

Chapter 6 Lecture - Chapter 6 Systems of Particles...

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1 Chapter 6 Systems of Particles Generaalized Eigenvalue Equaton We write the multiparticle Hamiltonian H which operates on a multiparticle wave state Ψ yielding the total energy E of the system. H (1,2,3,4,5,. .. N ) ! " (1,2,3,4,. .. N ) = E ! (1,2,3,4,. .. N ) 1 # ( x 1 , p 1 ) . ...... N # ( x N , p N ) This equation can only be solved on computers if at all for multiparticle states, molecules,etc. Independent or Weakly interacting Particles If the particles are weakly interacting we might consider the Hamiltonian as a sum of single particle Hamiltonians where the potential is a mean field felt by all particles equally. H (1) + H (2) + H (3) + .... H ( N ) ( ) ! (1,2,3,. .. N ) = E 1 + E 2 + E 3 + ..... + E N ( ) ! (1,2,3,. .. N ) Perfectly Seperable equation so choose a combination wave function ! 1, 2, 3, 4,. .. N = # 1 (1) 1 (2) 1 (3) 1 (4). ... 1 ( N ) where we solve the system of sin gle particle equations H 1 (1) 1 = E 1 (1) 1 H 2 (2) 2 = E 2 (2) 2 H N N = E N N In principle we can solve one equation and have all the solutions. Example: Let a system of 4 particles be confined to an infinite square well of width a . ! n ( i ) = 2 a sin( n x a ) E n = ! 2 2 m n a # $ % ( 2 ground state ) (1,2,3,4) 1,1,1,1 = n = 1 (1) n = 1 (2) n = 1 (3) n = 1 (4) E = 4 E 1 1 2 3 4 6 5 N n=1 n=2
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2 Distinquishable Particles Classically we are able to label each particle and then in principle follow there time development.
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This note was uploaded on 09/13/2009 for the course PHYS 451 taught by Professor Cavaglia,m during the Fall '08 term at Ole Miss.

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Chapter 6 Lecture - Chapter 6 Systems of Particles...

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