Computational Physics

Exercises 83

The bosons interact via aδ-function potential:

H=−

∑N

i= 1

∂^2

∂xi^2

+g

∑N

i>j

δ(xi−xj).

This means that the particles feel each other only if they are at the same position: they

experience an infinitely large attraction with an infinitely short range. Although this

problem can be solved exactly, we consider the Hartree–Fock approximation here.

A boson wave functionis symmetric under particle exchange. This means that

allthe particles are in the same orbitalφ.

(a) Show that the kinetic term of the Hamiltonian has the form:

〈



∣∣

∣∣

∣

−

∑N

i= 1

∂^2

∂x^2 i

∣∣

∣∣

∣



〉

=−N

〈

φ

∣∣

∣∣d

2

dx^2

∣∣

∣∣φ

〉

.

(b) Show that the expectation value of the interaction potential is given by

〈



∣∣

∣∣

∣∣g

∑

i<j

δ(xi−xj)

∣∣

∣∣

∣∣

〉

=

1

2

N(N− 1 )

∫

dx|φ(x)|^4.

(c) Show, by minimisation of the energy-functional

〈E〉=

〈|H|〉

〈|〉

with respect toφ, that the Hartree–Fock equation reads

[

−

∂^2

∂x^2

−g(N− 1 )|φ(x)|^2

]

φ(x)=φ(x).

(d) The solution to this last equation is found as

φ(x)=

[

( 1 / 8 )g(N− 1 )

] 1 / 2

cosh

[

( 1 / 4 )g(N− 1 )x

];

=

1

16

g^2 (N− 1 )^2.

Show that this function indeed satisfies the Hartree–Fock equation.

4.6 Consider the Fock spectrum of a many-electron system. In the ground state, theN
electrons fill the lowestNlevels of this spectrum. Consider the same spectrum, but
now with one electron removed from it. This means that the system has been
ionised. Show from the expressions for the Fock operator and the energy in terms of
the spin-orbitals,Eqs. (4.41)and(4.48), that the ionisation energy is equal to the
difference between the sum over the occupied Fock levelskin the ground state and
the same sum for the ionised state.
It is then clear that the same holds for adding an electron to the ground state, and
therefore for moving an electron from levelato levelb(by first removing the
electron from levelaand then adding one in levelb).