Download Pauli Exclusion Principle

Quantum Theory
But if an electron acts as a wave when it is
moving, WHAT IS WAVING?
When light acts as a wave when it is moving,
we have identified the
ELECTROMAGNETIC FIELD
as waving.
But try to recall: what is the electric field?
Can we directly measure it?
Quantum Theory
Recall that by definition, E = F/q. We can
only determine that a field exists by
measuring an electric force! We have
become so used to working with the electric
and magnetic fields, that we tend to take
their existence for granted. They certainly
are a useful construct even if they don’t
exist.
Quantum Theory
We have four LAWS governing the electric and
magnetic fields that as a group are called:
MAXWELL’S EQUATIONS (Gauss’ Law for electric
fields which is equivalent to Coulomb’s Law; Gauss’ Law for magnetic
fields, Ampere’s Law which is equivalent to the magnetic force law,
and Faraday’s Law).
By combining these laws we can
get a WAVE EQUATION for E&M fields, and
from this wave equation we can get the speed of
the E&M wave and even more (such as reflection
coefficients, etc.).
Quantum Theory
But what do we do for electron waves?
What laws or new law can we find that will
work for matter to give us the wealth of
predictive power that MAXWELL’S
EQUATIONS have given us for light?
Quantum Theory
The way you get a law is to try to explain
something you already know about, and
then see if you can generalize. A successful
law will explain what you already know
about, and predict things to look for that
you may not know about. This is where the
validity (or at least usefulness) of the law can
be confirmed.
Quantum Theory
Schrodinger started with the idea of
Conservation of Energy: KE + PE = Etotal .
He noted that there are two relations between the
particle and wave ideas: E=hf and p=h/ ,
and both could be related to energy:
• KE = (1/2)mv2 = p2/2m, and that =h/p, so that
p = h/ = (h/2)*(2/) = k = p, so
KE = 2k2/2m
• Etotal = hf = (h/2)*(2f) = .
Quantum Theory
He then took a nice sine wave, (actually a cosine wave
which differs from a sine wave by a phase of 90o) and
called whatever was waving, :
(x,t) = A cos(kx-t) = Real part of Aei(kx-t).
He noted that both k and  were in the
exponent, and could be obtained by
differentiating. So he tried operators:
Quantum Theory
(x,t) = A cos(kx-t) = Aei(kx-t) .
pop = -i[d/dx] = -i[ikAei(kx-t)] = k
= (h/2)*(2/)* = (h/ = p .
Similary:
Eop= i[d/dt] = i[-iAei(kx-t)] = 
= (h/2)*(2f)* = (hf = E .
(Note the use of imaginary numbers in real physics!)
Quantum Theory
Conservation of Energy: KE + PE = Etotal
becomes with the use of the momentum and
energy operators:
-(2/2m)*(d2/dx2) + PE* = i(d/dt)
which is called SCHRODINGER’S
EQUATION. If it works for more than the
free electron (where PE=0), then we can
call it a LAW.
Quantum Theory
What is waving here? 
What do we call ? the wavefunction
Schrodinger’s Equation allows us to solve
for the wavefunction.
The operators then allow us to find out
information about the electron, such as its
energy and its momentum.
Quantum Theory
To get a better handle on , let’s consider
light: how did the E&M wave relate to the
photon?
Quantum Theory
The photon was the basic unit of energy for the light.
How did the field strength relate to the energy?
[Recall energy in capacitor, Energy = (1/2)CV2, where
Efield = V/d, and for parallel plates C// = KoA/d, so:
Energy = (1/2)CV2 = (1/2)*(KoA/d)*(Efieldd)2
= KoEfield2 * Vol, or Energy/Vol Efield2.]
The energy per volume in the wave depended on
the field strength squared.
Quantum Theory
Since Energy density is proportional to field
strength squared, AND energy density is
proportional to the number of photons per
volume, THEN that implies that the
number of photons is proportional to the
square of the field strength.
This then can be interpreted to mean that the
square of the field strength at a location
is related to the probability of finding a
photon at that location.
Quantum Theory
In the same way, the square of the
wavefunction is related to the probability of
finding the electron!
Since the wavefunction is a function of both x
and t, the probability of finding the electron
is also a function of x and t!
Prob(x,t) = (x,t)2
Quantum Theory
Different situations for the electron, like being
in the hydrogen atom, will show up in
Schrodinger’s Equation in the potential
energy (PE) part.
Different PE functions (like PE = -ke2/r for the
hydrogen atom) will cause the solution of
Schrodinger’s equation to be different, just
like different PE functions in the normal
Conservation of Energy will cause different
speeds to result for the particles.
Quantum Theory
When we solve the Schrodinger Equation using the
potential energy for an electron around a proton
(hydrogen atom), we get a 3-D solution that gives us
three quantum numbers: one for energy (n), one for
angular momentum (L), and one for the z component
of angular momentum (mL). Thus the normal quantum
numbers come out of the theory instead of being put
into the theory as Bohr had to do.
There is a fourth quantum number (recall your
chemistry), but we need a relativistic solution to the
problem to get that fourth quantum number (spin).
Extending the Theory
How do we extend the quantum theory to
systems beyond the hydrogen atom?
For systems of 2 electrons, we simply have a
 that depends on time and the coordinates
of each of the two electrons:
(x1,y1,z1,x2,y2,z2,t)
and the Schrodinger’s equation has two
kinetic energies instead of one.
Extending the Theory
It turns out that the Schodinger’s Equation can
be separated (a mathematical property):
(x1,y1,z1,x2,y2,z2,t)
= Xa(x1,y1,z1) * Xb(x2,y2,z2) * T(t) .
This is like having electron #1 in state a, and
having electron #2 in state b. Note that
each state (a or b) has its own particular set
of quantum numbers.
Extending the Theory
However, from the Heisenberg Uncertainty
Principle (i.e., from wave/particle duality), we
are not really sure which electron is electron
number #1 and which is number #2. This
means that the wavefunction must also
reflect this uncertainty.
Extending the Theory
There are two ways of making the
wavefunction reflect the indistinguishability
of the two electrons: (symmetric)
sym = [Xa(r1)*Xb(r2) + Xb(r1)*Xa(r2) ]* T(t)
and (antisymmetric)
anti = [Xa(r1)*Xb(r2) - Xb(r1)*Xa(r2) ]* T(t)
.
[We don’t have to worry about  being negative,
since the probability (which must be positive)
depends on 2 .]
Extending the Theory
sym = [Xa(r1)*Xb(r2) + Xb(r1)*Xa(r2) ]* T(t)
anti = [Xa(r1)*Xb(r2) - Xb(r1)*Xa(r2) ]* T(t)
.
Which (if either) possibility agrees with
experiment?
It turns out that some particles are explained
nicely by the symmetric, and some are
explained by the antisymmetric.
BOSONS
sym = [Xa(r1)*Xb(r2) + Xb(r1)*Xa(r2) ]* T(t)
Those particles that work with the symmetric
form are called BOSONS. All of these
particles have integer spin as well. Note
that if boson #1 and boson #2 both have the
same state (a=b), then  > 0. This means
that both particles CAN be in the same state
at the same location at the same time.
FERMIONS
anti = [Xa(r1)*Xb(r2) - Xb(r1)*Xa(r2) ]* T(t)
Those particles that work with the antisymmetric wavefunction are called
FERMIONS. All of these particles have
half-integer spin. Note that if fermion #1
and fermion #2 both have the same state,
(a=b), then  = 0. This means that both
particles can NOT be in the same state at
the same location at the same time.
Extending the Theory
BOSONS. Photons and alpha particles (2
neutrons + 2 protons) are bosons. These
particles can be in the same location with
the same energy state at the same time.
This occurs in a laser beam, where all the
photons are at the same energy
(monochromatic).
Pauli Exclusion Principle
FERMIONS. Electrons, protons and
neutrons are fermions. These particles can
NOT be in the same location with the same
energy state at the same time.
This means that two electrons going around
the same nucleus can NOT both be in the
exact same state at the same time! This is
known as the Pauli Exclusion Principle!
Pauli Exclusion Principle
Since no two electrons can be in the same
energy state in the same atom at the same
time, the concept of filling shells and
valence electrons can be explained and this
law of nature makes chemistry possible
(and so makes biology, psychology, sociology,
politics, and religion possible also)!
Thus, the possibility of chemistry is explained
by the wave/particle duality of light and
matter with electrons acting as fermions!