Direct Energy, 2018a

13 THOMAS FERMI ANALYSIS 299
Remember that E represents energy while −→ E represents electric eld. Electric
eld is the negative gradient of the voltage V (r).
−→ −→ E = −∇V. (13.25)
We can combine these expressions and Eq. 13.13 to write the rst term of
the Hamiltonian and the Lagrangian in terms of the generalized path.
E Coulomb e nucl
+ E e e interact
= 1 ∣ ∣∣
V
V 2 ɛ −→
∣ ∣∣
2
∇V (13.26)
(
H r, V, dV ) ( 1
∣ ∣∣
=
dr 2 ɛ −→
∣ ) ∣∣
2
∇V + E kinetic e
(13.27)
V
(
L r, V, dV ) ( 1
∣ ∣∣
=
dr 2 ɛ −→
∣ ) ∣∣
2
∇V − E kinetic e
(13.28)
V
The next task is to describe the remaining term E kinetic e
as a function
V
of the generalized path too. This task is a bit more challenging. We
continue to take the approach of making severe approximations untilit is
manageable. We need to express ρ ch (r) as a function of V (r). Then with
some algebra, E kinetic e
can be written purely as a function of V (r).
V
We want to generalize about the kinetic energy of the electrons. However,
each electron has its own velocity −→ v and momentum −→ M. These
quantities depend on position
−→ r = râr + θâ θ + φâ φ (13.29)
in some unknown way. Furthermore, the calculation of E kinetiic e
depends
V
on charge density ρ ch (r), which is the unknown quantity we are trying to
nd. We have more luck by describing these quantities in reciprocal space,
introduced in Sec. 6.4. Position is denoted in reciprocalspace by a wave
vector
−→ k =˜râr + ˜θâ θ + ˜φâ φ . (13.30)
We can describe the properties of a materialby describing how they
vary with position in realspace. For example, ρ ch (r) represents the charge
density of electrons as a function of distance r from the center of the atom.
We may be interested in how other quantities, such as the energy required
to rip o an electron or the kinetic energy internal to an electron, vary with
position in realspace too. Instead of describing how quantities vary with
position in realspace, we can describe how quantities vary with spatial
frequency of electrons. This is the idea behind representing quantities
in reciprocalspace. We may be interested in how the charge density of