Direct Energy, 2018a

13 THOMAS FERMI ANALYSIS 309
functional theory calculations do not make as many or as severe of assumptions
as were made above, especially for the E e e interact term. These calculations
have been used to calculate the angular dependence of the charge
density, the allowed energy states of electrons that are part of molecules,
the voltage felt by electrons at temperatures above absolute zero [136], and
many other microscopic properties of atoms. Density functional theory is
an active area of research. Often charge density is chosen as the generalized
path insteadof voltage [136].
Both Thomas [173] andFermi [174] includednumerical simulations.
Amazingly, these calculations were performedway before computers were
available! More recently, researchers have developed software packages for
applying density functional theory to calculate the allowed energy levels,
charge density, andso on of electrons aroundatoms andmolecules [178]
[179]. Because of the complexity of the calculations, parallel processing is
used. Computers with multiple processors, supercomputers, and graphics
cards with dozens of processors have all been used.
13.6 Problems
13.1. Generalizedmomentum is denedas
M =
∂L
∂ ( ).
dV
dr
(a) Findthe generalizedmomentum for the system describedby the
Lagrangian of Eq. 13.67.
(b) The generalizedmomentum does not have the units of momentum.
Identify the units of this generalized momentum.
(c) Write the Hamiltonian of Eq. 13.66 as a function of r, V, and
M but not as a function of dV . dr
(d) Write the Lagrangian of Eq. 13.67 as a function of r, V, and M
but not as a function of dV . dr
(e) Show that the Hamiltonian andLagrangian foundabove satisfy
the equation H = M dV −L. dr
13.2. In the analysis of this chapter, the generalizedpath was chosen as V
andthe generalizedpotential was chosen as ρ ch . The opposite choice
is also possible where the generalizedpath is ρ ch andthe generalized
potential is V .