385lec16 - PHYS 385 Lecture 16 - Many particle systems...

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PHYS 385 Lecture 16 - Many particle systems 16 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 16 - Many particle systems What's important : many-particle wavefunctions bosons and fermions symmetry of wavefunction Text : Gasiorowicz, Chap. 8 So far, we have discussed the motion of a single particle in an external fixed potential; we have not considered how the potential arises or that it might be associated with another particle. Let's now generalize our system to include the motion of several particles in interaction with each other. We won't actually implement most of the concepts presented here until much later in the course, but there are some issues that will arise in the next several lectures that are best dealt with now: many-particle Hamiltonians and wavefunctions reduced mass for two-body systems bosons and fermions symmetrization of the wavefunction. Many-particle Hamiltonians and wavefunctions Still working in one dimension, we consider a group of N particles having positions x 1 ... x N . The classical Hamiltonian for this system is H = Σ i ( p i 2 /2 m ) + V ( x 1 ... x N ), (1) where we take the potential energy to be independent of time. The usual operator replacement can be made for the momentum of each particle, yielding H = - h 2 1 2 m i i 2 x i 2 + V ( x 1 , x 2 ... x N ) (2) where the particles may each have a different mass according to m i . Because the coordinates of each particle are independent, the position-momentum commutator has the general form [ p i , x j ] = - i h ij . (3) The total momentum of the system has the same formal expression as in classical mechanics, P total = Σ i p i and is conserved, as estalbished in Gasiorowicz. The system of particles is described by the wavefunction ( x 1 ... x N ; t ) (4)
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PHYS 385 Lecture 16 - Many particle systems 16 - 2 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. The quantity |
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385lec16 - PHYS 385 Lecture 16 - Many particle systems...

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