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

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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 ) , ( ) 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 = = = = 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 = + = 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 + + = + + + + + = 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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08many - P460 - many particles 1 Many Particle Systems can...

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