Evaluation
Evaluation of expectation value of total energy (continue from last)
We visited once more the "counting of terms" in the
evaluation of expectation value of total energy;
It will be used in a variational method (as in the Schrödinger
equation from variation at the end of
(2012.10.11) last lecture note
)
1_pair_interaction_nonzero_matrix_elements.png
1_pair_interaction_nonzero_matrix_elements.png
This total energy will be used in a variational method
(as in the Schrödinger equation from variation at the end
of (2012.10.11)
last lecture note )
Lagrange multipliers
2_Variation_as_in_schroedinger.png
2_Variation_as_in_schroedinger.png
Performing, or evaluating "the N variations; with N
conditions; first for the "Hartree-only, disregarding exchange term"
WE SEE THAT WE RE_DERIVE THE HARTREE METHOD FRO VARIATIONAL APPROACH
(but the lowest part - the exchange term - see below)
3_exchange_term_NONLOCAL.png
3_exchange_term_NONLOCAL.png
In the lowest part of the above picture we see a "strange bahaviour"
emerging
from the exchange term. It is not like the "density" originating
from the direct term
Non-local interactions; in QM, all interactions could be
non-local
- see the delta-function which must be assumed to get the local
operations
- the only type known in classical mechanics (for particles of any
kind)
4_non-local-interaction.png
4_non-local-interaction.png
Here we correct some missing stars why re-viewing the
Hartree-Fock Equations
5_direct_and_exchange_potential.png
5_direct_and_exchange_potential.png
Non-local interactions are sometimes called "velocity-dependent
potentials".
Here is illustrted why.
6_velocity_dependent_non-local-Density_functional.png
6_velocity_dependent_non-local-Density_functional.png