1549380323-Statistical Mechanics Theory and Molecular Simulation

General formulation 411

periodicity of the box. These can be collected into a vectorni= (nx,i,ny,i,nz,i) of
integers, which leads to the following solution to eqn. (11.2.6):

ψni(ri) =

(

1

L

) 3 / 2

exp(2πinx,ixi/L) exp(2πiny,iyi/L) exp(2πinz,izi/L)

=

1

√

V

exp(2πini·ri/L). (11.2.8)

Similarly, each component of momentum is quantized, so that the momentum eigen-
values can be expressed as

pni=

2 π ̄h

L

ni, (11.2.9)

and the energy eigenvalues in eqn. (11.2.6) are just sums of the energies in eqn. (9.3.11)
overx,y, andz:

εni=

p^2 ni

2 m

=

2 π^2 ̄h^2

mL^2

|ni|^2. (11.2.10)

Multiplying the functions in eqn. (11.2.8) by spin eigenfunctions, the complete single-
particle eigenfunctions become

〈xi|nimi〉=φnimi(xi) =

1

√

V

e^2 πini·ri/Lχmi(si), (11.2.11)

and the total energy eigenvalues are given by a sum over single-particle eigenvalues

En 1 ,...,nN=

∑N

i=1

2 π^2 ̄h^2

mL^2

|ni|^2. (11.2.12)

Finally, since the eigenvalue problem is separable, complete fermionic and bosonic
wave functions can be constructed as follows. Begin by constructing a matrix

M =

      

φn 1 ,m 1 (x 1 ) φn 2 ,m 2 (x 1 ) ··· φnN,mN(x 1 )

φn 1 ,m 1 (x 2 ) φn 2 ,m 2 (x 2 ) ··· φnN,mN(x 2 )

· · ··· ·

· · ··· ·

· · ··· ·

φn 1 ,m 1 (xN) φn 2 ,m 2 (xN) ··· φnN,mN(xN)

      

. (11.2.13)

The properly symmetrized fermionic and bosonic wave functions areultimately given
by

Ψ(F)n 1 ,m 1 ,...,nN,mN(x 1 ,...,xN) = det(M)

Ψ(B)n 1 ,m 1 ,...,nN,mN(x 1 ,...,xN) = perm(M), (11.2.14)

where det and perm refer to the determinant and permanent of M,respectively. (The
permanent of a matrix is just determinant in which all the minus signs are changed