1549380323-Statistical Mechanics Theory and Molecular Simulation

390 Quantum mechanics

in creating the given permutation, the wave function will pick up a factor of−1 for
eachexchange of two particles that is performed:

ΨF(x 1 ,...,xN) = (−1)NexΨF(xP(1),....,xP(N)), (9.4.13)

whereNexis the total number of exchanges of two particles required in orderto
achieve the permutationP(1),...,P(N). AnN-particle bosonic or fermionic state can
be created from a state Φ(x 1 ,...,xN) which is not properly symmetrized but which,
nevertheless, is an eigenfunction of the Hamiltonian

HˆΦ =EΦ. (9.4.14)

Since there areN! possible permutations of theNparticle labels in anN-particle
state, the bosonic state ΨB(x 1 ,...,xN) is created from Φ(x 1 ,...,xN) according to

ΨB(x 1 ,...,xN) =

1

√

N!

∑N!

α=1

PˆαΦ(x 1 ,...,xN), (9.4.15)

wherePˆαcreates 1 of theN! possible permutations of the indices. The fermionic state
is created from

ΨF(x 1 ,...,xN) =

1

√

N!

∑N!

α=1

(−1)Nex(α)PˆαΦ(x 1 ,...,xN), (9.4.16)

whereNex(α) is the number of exchanges needed to create permutationα. TheN! that
appears in the physical states is exactly theN! introducedad hocin the expressions
for the classical partition functions to account for the identical nature of the particles
not explicitly treated in classical mechanics.

9.5 Problems

9.1. Generalize the proof in Section 9.2.3 of orthogonality of the eigenvectors of a

Hermitian operator to the case that some of the eigenvalues of theoperator are

degenerate. Start by considering two degenerate eigenvectors: If|aj〉and|ak〉

are two eigenvectors ofAˆwith eigenvalueaj, show that two new eigenvectors

|a′j〉and|a′k〉can be constructed such that|a′j〉=|aj〉and|a′k〉=|ak〉+c|aj〉,

wherecis a constant, and determinecsuch that〈a′j|a′k〉= 0. Generalize the

procedure to an arbitrary degeneracy.

9.2. A spin-1/2 particle that is fixed in space interacts with a uniform magnetic

fieldB. The magnetic field lies entirely along thez-axis, so thatB= (0, 0 ,B).

The Hamiltonian for this system is therefore

Hˆ=−γBSˆz.

The dimensionality of the Hilbert space for this problem is 2.