1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 393

a. If the Hamiltonianˆh(ˆx,ˆp) is of the form

ˆh(ˆx,pˆ) = pˆ

2

2 m

+U(ˆx),

show that the eigenvalue problem forHˆ can be expressed asN single-

particleeigenvalue problems of the form

[

−

̄h^2

2 m

∂^2

∂x^2

+U(x)

]

ψki(x) =εkiψki(x),

such that theN-particle eigenvaluesEk 1 ,...,kN, which are characterized

byNquantum numbers, are given by

Ek 1 ,...,kN=

∑N

i=1

εki.

b. Show that if the particles could be treated as distinguishable, then the
eigenfunctions ofHˆcould be expressed as a product

Φk 1 ,...kN(x 1 ,...,xN) =

∏N

i=1

ψki(xi).

c. Show that if the particles are identical fermions, then the application of

eqn. (9.4.16) leads to a set of eigenfunctions Ψk 1 ,...,kN(x 1 ,...,xN) that is

expressible as the determinant of a matrix whose rows are of the form

ψk 1 (xP(1)) ψk 2 (xP(2))···ψkN(xP(N)).

Recall thatP(1),...,P(N) is one of theN! permutations of the indices

1,...,N. Give the general form of this determinant. (This determinant is

called aSlater determinantafter its inventor John C. Slater (1900–1976).)

Hint: Try it first forN= 2 andN= 3.

d. Show that if the particles are bosons rather than fermions, then the eigen-
functions are exactly the same as those of part c except for a replacement
of the determinant by a permanent.

Hint: The permanent of a matrix can be generated from the determinant

by replacing all of the minus signs with plus signs. Thus, for a 2×2 matrix

M=

(

a b

c d

)

,

perm(M) =ad+bc.