This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...
Now that we have treated the Hydrogen like atoms in some detail, we now
proceed to discuss the nextsimplest system: the Helium atom. In this
situation, we have tow electrons – with coordinates
z r1 and r2 – orbiting a nucleus with charge Z = 2
located at the point R. Now, for the hydrogen atom
we were able to ignore the motion of the nucleus
r2 by transforming to the center of mass. We then
R obtained a Schrödinger equation for a single
y
effective particle – with a reduced mass that was
very close to the electron mass – orbiting the
x origin. It turns out to be fairly difficult to
r1
transform to the center of mass when dealing with
three particles, as is the case for Helium. However, because the nucleus is
much more massive than either of the two
electrons (MNuc ≈ 7000 mel) it is a very good z
approximation to assume that the nucleus sits at
the center of mass of the atom. In this
approximate set of COM coordinates, then, R=0 r2
and the electron coordinates r1 and r2 measure
y
the between each electron and the nucleus.
Further, we feel justified in separating the
motion of the nucleus (which will roughly x
correspond to rigidly translating the COM of the
r1
atom) from the relative d the electrons orbiting
the nucleus within the COM frame. Thus, in what follows, we focus only on the
motion of the electrons and ignore the motion of the nucleus.
We will treat the quantum mechanics of multiple particles (1,2,3…) in much the
same way as we described multiple dimensions. We will invent operators r̂1 , r̂2 ,
r̂3 , … and associated momentum operators p̂1 , p̂ 2 , p̂3 …. The operators for a
given particle (i) will be assumed to commute with all operators associated with
any other particle (j):
[r̂1, p̂ 2 ] = [ p̂2 , r̂3 ] = [r̂2 , r̂3 ] = [ p̂1, p̂3 ] = ... ≡ 0
, 5.61 Physical Chemistry 25 Helium Atom page 2
Meanwhile, operators belonging to the same particle will obey the normal
commutation relations. Position and momentum along a given axis do not
commute:
⎡ x̂1, p̂x ⎤ = i� ⎡ ŷ , p̂ ⎤ = i� ⎡ ẑ , p̂ ⎤ = i�
⎣ 1⎦ ⎣ 1 y1 ⎦ ⎣ 1 z1 ⎦
while all components belonging to different axes commute:
xˆ1 yˆ1 = yˆ1 xˆ1 p̂z1 yˆ1 = yˆ1 pˆ z1 pˆ z1 p̂x1 = pˆ x1 pˆ z1 etc.
As you can already see, one of the biggest challenges to treating multiple electrons
is the explosion in the number of variables required!
In terms of these operators, we can quickly write down the Hamiltonian for the
Helium atom:
Kinetic Energy NucleusElectron 1
Of Electron 1 ElectronElectron
Attraction Repulsion
This Hamiltonian looks very intimidating, mainly because of all the constants (e, me,
ε0, etc.) that appear in the equation. These constants result from our decision to
use SI units (meter, gram, second) to express our lengths, masses and energies.
This approach is quite awkward when the typical mass we’re dealing with is 1028
grams, the typical distance is 1010 meters and the typical energy unit is 1018
Joules. It is therefore much simpler to work everything out in what are called
atomic units. In this system of units we choose our unit of mass to be equal to the
electron mass, me, our unit of charge to be equal to the electron charge, e, and our
unit of angular momentum to be � . Further, we choose to work in electrostatic
units, so that the permittivity of free space (4π ε0) is also 1. The result of these
four choices is twofold. First of all, the Hamiltonian of the Helium atom (and
indeed of any atom or molecule) simplifies greatly because all the constants are
unity and can be omitted in writing the equations:
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