# 08many - P460 many particles 1 Many Particle Systems •...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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:...
View Full Document

## 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.

### Page1 / 12

08many - P460 many particles 1 Many Particle Systems •...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online