IdenticalParticlesRevisited

IdenticalParticlesRevisited - Identical Particles Revisited...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Identical Particles Revisited Michael Fowler Introduction For two identical particles confined to a one-dimensional box, we established earlier that the normalized two-particle wavefunction ( ) 12 , x x ψ , which gives the probability of finding simultaneously one particle in an infinitesimal length dx 1 at x 1 and another in dx 2 at x 2 as () 2 , x xd x d x , only makes sense if 2 21 , 2 , x xx ψψ = x , since we don’t know which of the two indistinguishable particles we are finding where. It follows from this that there are two possible wave function symmetries: ( ) ( ) ,, x x = or ( )( , ) , x x =− . It turns out that if two identical particles have a symmetric wave function in some state, particles of that type always have symmetric wave functions, and are called bosons . (If in some other state they had an antisymmetric wave function, then a linear superposition of those states would be neither symmetric nor antisymmetric, and so could not satisfy 2 , 2 , x = x ) , .) Similarly, particles having antisymmetric wave functions are called fermions . (Actually, we could in principle have ( , i x xe x x α = , with α a constant phase, but then we wouldn’t get back to the original wave function on exchanging the particles twice. Some two-dimensional theories used to describe the quantum Hall effect do in fact have excitations of this kind, called anyons , but all ordinary particles are bosons or fermions.) To construct wave functions for three or more fermions, we assume first that the fermions do not interact with each other, and are confined by a spin-independent potential, such as the Coulomb field of a nucleus. The Hamiltonian will then be symmetric in the fermion variables, ( ) ( ) ( ) 222 123 1 2 3 /2 H p mp mp m Vr Vr Vr =+++ + + + + GGG G G G …… and the solutions of the Schrödinger equation are products of eigenfunctions of the single- particle Hamiltonian ( ) 2 Hp mV r =+ GG , . However, these products, for example () ( ) () ab c do not have the required antisymmetry property. Here a , b , c , … label the single-particle eigenstates, and 1, 2, 3, … denote both space and spin coordinates of single particles, so 1 stands for ( . The necessary antisymmetrization for the particles 1, 2 is achieved by subtracting the same product wave function with the particles 1 and 2 interchanged, so ) 11 , rs G c is replaced by ( ) ( ) ( ) ( ) ( ) ( ) 213 c a bc , ignoring overall normalization for now. But of course the wave function needs to be antisymmetrized with respect to all possible particle exchanges, so for 3 particles we must add together all 3! permutations of 1, 2, 3 in the state a , b , c , with a factor - 1 for each particle exchange necessary to get to a particular ordering from the original ordering of 1 in a , 2 in b , and 3 in c . In fact, such a sum over permutations is precisely the definition of the determinant, so, with the appropriate normalization factor:
Background image of page 1

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

View Full DocumentRight Arrow Icon
2 () ( ) ( ) ( ) () () () 11 1 1, 2, 3 2 2 2 3!
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

Page1 / 6

IdenticalParticlesRevisited - Identical Particles Revisited...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online