Helium Atom - Two electron systems
For helium, we have three particles, in hydrogen we had two. As
in hydrogen, we eliminate
the center of mass, so that we are left basically with two
particles (CM we place on the nucleus,
corrections are very small and here we do not consider them - reduced
mass is the keyword)
Instead of one position vector - 3 coordinates - we have now two
vectors - 6 coordinates.
And we have all double - but in addition the electron
repulsion
1.H_vs_He_6_coordinates.png
1.H_vs_He_6_coordinates.png
Separation of variables - the radial wavefunction and the angular
part. We could also separate
the two vectors - one for each electron - if they would be
separable. Let us look at the separability
Here we discuss separation of variables in general - the
condition is that the differential equations
contain only terms which are constructed from each of the variables
separately. And the total
operations are expressed as a sum of these terms. Then the
solutions can be written (assumed)
as products of two unknown functions and the equations can be
separated.
Condition: LINEAR OPERATIONS
2.Separation_of_variables.png
2.Separation_of_variables.png
What are expansions in a set of functions? Known from many areas
of mathematics, physics, oceanography, ......
MOST OF US KNOW THE FOURIER SERIES METHOD
3.Basis_functions.png
3.Basis_functions.png
EXAMPLE: Fourier Series, spherical harmonics; Physics of drumming
Product wavefunction (separability) - and product wavefunctions
as indication of INDEPENDENT MOTION.
CAN BE TRACED TO PROBABILITY OF INDEPENDENT EVENTS - PRODUCT OF
PROBABILITIES
Probability density - products of probability densities (think about
it - dV in 3-dim goes to dW in 6-dim,
so that indeed the density for 2-paricle system has the dimension
"Length to the power of -6"
4._Separable_to_Independent.png
4._Separable_to_Independent.png
similarly in classical physics - there we can also separate even the
newton's equations.
Example: Kepler problem, the radial motion and the angular motion
can be treated separately.
Symmetry an antisymmetry is often a starting point in textbooks.
We review this point,
but below show that this consideration is in fact made necessary by
the observation-based
Pauli exclusion principle (postulated by Pauli). Concept of
independent particles
followed from the Pauli principle, see below.
5._Exclusion_Pauli_Antisymmetry.png
5._Exclusion_Pauli_Antisymmetry.png
Aufbau - Build-up principle. Why do not all the electrons in
complex atoms
stay crowded in the lowest state? Well, Pauli explained it by
postulating the
exclusion principle.
Identical particles was then a simpler and more powerful concept -
and lead to
the generalization - and explanation of Fermi and Bose-Einstein
statistics.
6._Pauli_Fermi_Aufbau_SPIN.png
6._Pauli_Fermi_Aufbau_SPIN.png
Next time: spin; 2 particles with spin.