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Unformatted text preview: P460  many particles 1 Many Particle Systems • can write down the Schrodinger Equation for a many particle system • with x i being the coordinate of particle i (r if 3D) • the Hamiltonian has kinetic and potential energy • if only two particles and V just depends on separation then can treat as “one” particle and use reduced mass (ala classical mech. or H atom) • in QM, H does not depend on the labeling. And so if any i b j and j b i, you get the same observables or state this as (for 2 particles) H(1,2)=H(2,1) t x x x i x x x H n n ∂ ∂ = ) , ( ) , ( 2 1 2 1 K h K ψ ψ ) , ( ) 2 1 2 1 ( 2 1 2 2 2 2 2 1 2 n x x x V x m x m H K K h + ∂ ∂ + ∂ ∂ − = P460  many particles 2 Exchange Operator • for now keep using 2 particle as example. Use 1,2 for both space coordinates and quantum states (like spin) • can formally define the exchange operator • as the eigenvalues of H do not depend on 1,2, implies that P 12 is a constant of motion • can then define symmetric and antisymmetric states. If start out in an eigenstate, then stays in it at future times N = normalization. so for 2 and 3 particle systems 1 : ) 2 , 1 ( ) 1 , 2 ( )) 2 , 1 ( ( ) 1 , 2 ( ) 2 , 1 ( 12 12 12 12 ± = ⇒ = = = λ α α α α α s eigenvalue u u P u P P u u P [ ] , 12 12 = ∂ ∂ ⇒ = t P P H )) 1 , 2 ( ) 2 , 1 ( ( 1 )) 1 , 2 ( ) 2 , 1 ( ( 1 ψ ψ ψ ψ ψ ψ − = + = N N A S )) 2 , 3 , 1 ( ) 2 , 1 , 3 ( ) 1 , 2 , 3 ( ) 1 , 3 , 2 ( ) 3 , 1 , 2 ( ) 3 , 2 , 1 ( ( 1 )) 2 , 3 , 1 ( ) 2 , 1 , 3 ( ) 1 , 2 , 3 ( ) 1 , 3 , 2 ( ) 3 , 1 , 2 ( ) 3 , 2 , 1 ( ( 1 ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ − + − + − = + + + + + = N N A S P460  many particles 3 Schrod. Eq. For 2 Particles • have kinetic energy term for both electrons (1+2) • let V 12 be 0 for now (can still make sym/antisym in any case) • easy to show then that one can then separate variables and the wavefunction is:...
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.
 Spring '08
 Johnson,C
 Energy, Potential Energy, Quantum Physics, Schrodinger Equation

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