Direct Energy, 2018a

296 13.3 Derivation of the Lagrangian
13.2.3 Reciprocal Space Concepts
The idea of reciprocal space was introduced in Section 6.4 in the context
of crystalline materials. We can describe the location of atoms in a crystal,
for example, as a function of position where position −→ r is measured in
meters. In this chapter, we are interested in individual atoms instead of
crystals composed of many atoms. We can plot quantities like energy E( −→ r )
or voltage V ( −→ r ) as a function of position. Figure 6.11, for example, plots
energy versus position inside a diode. In Section 6.4, the idea of wave
vector −→ k in units of m −1 was introduced. Wave vector represents the
spatial frequency. We saw that we could plot energy or other quantities
as a function of wave vector, and Fig. 6.8 is an example of such a plot.
We will need the idea of wave vector in this chapter because we describe a
situation where we do not know how the energy varies with position, but
we do know something about how the energy varies with wave vector.
13.3 Derivation of the Lagrangian
The purpose of this chapter is to nd the voltage V (r) and the charge
density ρ ch (r) around an atom, and we will use calculus of variations to
accomplish this task. We need to make some rather severe assumptions to
make this problem manageable. Consider an isolated neutral atom with
many electrons around it. Assume T ≈ 0 K, so all electrons occupy the
lowest possible energy levels. Assume the atom is spherically symmetric.
All of the quantities we encounter, such as voltage, charge density, and
Lagrangian, vary with r but do not vary with θ or φ. We will use spherical
coordinates with the origin at the nucleus of the atom. While quantities
vary with position, assume no quantities vary with time. The charge density
ρ ch (r) tells us where the electrons are most likely on average to be found.
It is related to the quantummechanical wave function, ψ, by
ρ ch = −q ·|ψ| 2 (13.17)
where q is the magnitude of the charge of an electron. Assume that all of
the electrons surrounding the atomare distributed uniformly and can be
treated as if they were a uniformelectron cloud of some charge density.
Pick one of the electrons of the atom, and consider what happens when
the electron is moved radially in and out. Figure 13.1 illustrates this situation.
As the electron moves, energy conversion occurs. The goal of this
section is to write down the Hamiltonian and Lagrangian for this energy
conversion process. We write these quantities in the units of energy per
unit volume per valence electron under consideration.