Download Pauli Exclusion Principle

Quantum Mechanics and
Atomic Physics
Lecture 21:
Pauli Exclusion Principle and MultiMulti-electron atoms
http://www.physics.rutgers.edu/ugrad/361
http://www physics rutgers edu/ugrad/361
Prof. Sean Oh
Last time
: Fine structure constant
„
„
„
„
„
Electron mass : mc2 : ~0.5 MeV
Bohr
h energies:
i off order
d α2mc2 : ~100 eV
Fine structure: of order α4mc2 : ~10-4 eV
Lamb shift: of order α5mc2 : ~10-6 eV
Hyperfine splitting: of order (m/mp)α4mc2 :
~10-6 eV
Multi--Electron Atoms
Multi
„
Atoms with 2 or more electrons have
a new feature:
Electrons are indistinguishable!
g
„ There is no way to tell them apart!
„
„
Any measurable quantity (probability,
(probability
expectation value, etc.) must not
depend on which electron is labeled
1, 2, etc.
ΨA(1)
ΨB(2)
S.E. for MultiMulti-electron atoms
„
Let’s
Let s consider two electrons in Helium with
coordinates:
„
The total Hamiltonian operator for this system
is
1
„
2
So the Schrodinger equation is:
S.E. for MultiMulti-electron atoms
„
The total p
potential Vtot has 3 contributions:
1.
2.
3.
„
„
„
V between electron 1 and the nucleus
V between electron 2 and the nucleus
V between electron 1 and electron 2
For now, let’s consider only #1 and #2
S
So,
Note that the potential function is the same for both
electrons
S.E. for MultiMulti-electron atoms
„
„
„
„
We get the usual separation of variables
Each Ψ will depend on quantum numbers
n, l, ml, ms
So, A and B stand for the particular sets of quantum
numbers
So, let’s call ΨA(1) eigenfunction for electron #1 and
has the quantum numbers symbolized by A.
A
S.E. for MultiMulti-electron atoms
„
So, total eigenfunction solution is:
„
And with this separability assumption
assumption, the SS.E.
E
becomes:
„
This equation suggests that we write the total energy E
as:
S.E. for MultiMulti-electron atoms
„
So we can separate this neatly into two
So,
independent expressions:
„
((This can be extended to anyy number of nonnoninteracting particles!)
Exchange electrons
„
So, we said that the total eigenfunction is:
„
If we exchange electrons 1 and 2 we get:
„
But the first equation gives:
„
And the second (exchanged) equation gives:
„
But these two probabilities are different!
„
„
This is not acceptable!
p
This means that the expressions for Ψtot above are not valid
solutions!
Exchange electrons
„
Since:
„
We need this to be satisfied:
„
So the total eigenfunction:
„
„
+ sign: Symmetric eigenfunction ΨSymm
- sign:
g Anitsymmetric
y
eigenfunction
g
ΨAnti
Symmetric and Antisymmetric
„
ΨSymm
and ΨAnti
S
A i are degenerate!
„ Same energy
„ They exhibit “exchange degeneracy”
Th h
They
have the
h right
i h properties:
i
„
Note 1/√2 is for normalization, assuming that Ψtot is normalized:
„
Pauli Exclusion Principle
„
„
Principle was formulated by Wolfgang Pauli in
1925.
“Weak
Weak form
form”::
„
In an atom, no two electrons can be in the same
quantum state
state, i.e.
i e the same set of quantum
numbers: n, l, ml, ms
Pauli Exclusion Principle
„
Suppose electrons 1 and 2 are in the same quantum
state A.
A Then:
„
„
„
So, ΨSymm permits 2 electrons in the same state.
So, ΨSymm violates the Pauli exclusion principle
ΨAnti obeys
b the
th Pauli
P li exclusion
l i principle.
i i l
Pauli Exclusion Principle
„
“Strong” form of Pauli Exclusion Principle:
„
„
„
A multi
multi--electron system must have an antisymmetric total
eigenfunction.
i f
i
“Strong” because it also incorporates indistinguishability.
All particles of half
half--integer spin (1/2, 3/2, …) have
antisymmetric total eigenfunctions and are called “Fermions”,
obeying Fermi
Fermi--Dirac statistics
„
„
Electrons protons
Electrons,
protons, neutrons
All particles of integer spin (0, 1, 2, …) have symmetric total
eigenfunctions, and are called “Bosons”, obeying Bose
Bose-Einstein statistics.
statistics
„
Photons, alpha, W and Z particles
Required for Bosons
Ψtot ∝ ΨA (1)ΨB (2) ± ΨA (2)ΨB (1)
Required for Fermions
Helium Example
„
„
„
Normal Helium (4 2 He)
„ Even number of spin 1/2 constituents
„ 2 protons,
protons 2 neutrons and 2 electrons
„ Is a Boson
Helium--3 (3 2 He)
Helium
„ 2 protons, 1 neutrons, 2 electrons
„ Is
I a Fermion
F r i
Generally, Even number sum of
protons+neutrons+electrons Î Boson
Odd number sum Î Fermion
Total Fermion Eigenfunction
„
„
„
„
„
So,, for Fermion,, total eigenfunction
g
must be
antisymmetric
Can write:
So, Ψ(space) and Ψ(spin) must have opposite
symmetry in
i order
d ffor ΨAnti to be
b antisymmetric
i
i
We had used A and B as abbreviations for particular
sets of n,
n l, ml, ms
Now let’s use a and b as abbreviations for particular
sets of n, l, ml
„ i.e. just the space part
Space and Spin Eigenfunctions
„
„
The space wavefunctions are analogous to R
RΘΦ
ΘΦ for
Hydrogen
For spins
spins, there is no spatial wavefunction
wavefunction,, so
„ Symbolically,
Spin Eigenfunctions
„
„
„
„
Example:
But this does not have definite symmetry:
So the antisymmetric χ, corresponding to ΨSymm(space)
So,
(singlet state):
The symmetric χ, corresponding to ΨAnti(space); there
are three ways to do it (triplet state):
( +1/2,+1/2 )
( -1/2,-1/2 )
Spin Angular Momentum
„
In a two electron atom (Helium) the spin
angular
l rm
momentum
m t m off the
th two
t electrons
l tr
couple
pl
to give the total spin:
„
But what is s’?
Spin Angular Momentum
„
If s’=0, ms’ = 0 only, and this is the singlet state, which is
antisymmetric.
antisymmetric
„ We have opposite spins, and S’
S’=0
=0
„
If s’=1, ms’ = -1, 0, +1, and this is why we get the triplet state
which is symmetric.
„ We have
h
parallel
ll l spins
i
Parallel Spins
„
For parallel spins:
„
„
„
„
„
s 1
s’=1
triplet state in Helium (called Orthohelium
Orthohelium))
χ symmetric
Ψ(space) antisymmetric.
antisymmetric.
Suppose electrons get close, so a=b (same spatial
quantum numbers):
„
„
„
Low probability for electrons to have similar coordinates
Parallel--spin electron repel each other, over and above the
Parallel
Coulomb repulsion.
This “exchange” force mainly reflects the exclusion principle
Antiparallel Spins
„
For antiparallel spins:
„
„
„
„
s’=0
’ 0
Singlet state in Helium (called Parahelium)
Parahelium)
χ antisymmetric
y
Ψ(space) symmetric
„
Suppose electrons get close, so a=b (same spatial
quantum numbers):
„
Thi iis llarge, so antiparallelThis
antiparallel
ti
ll l-spin
i electrons
l t
attract
tt t each
h
other via the “exchange” force.
Example
„
Construct total (spatial+spin
(spatial+spin)) ground state
electron configuration of a Helium atom using
the spatial hydrogenic wavefunction notation of
Example
„
Construct total (spatial+spin
(spatial+spin)) first excited state
configuration of a Helium atom using the spatial
hydrogenic wavefunction
„
If you consider Coulomb repulsion between
electrons, will the first excited state be in a
singlet state ((parahelium
parahelium)) or in a triplet state
(orthohelium
orthohelium)?
)?
Example
„
If we put 5 electrons (fermions!) in an infinite square
well, what is the ground state energy?
„
„
„
Recall the energies for an infinite square well are:
2 2
π
h
n 2 π 2 h2
2
En =
= n E1 , where E1 =
2
2 L
2mL
2 L2
2mL
Since electrons are fermions, 2 electrons will populate n=1,
2 in n=2,, and 1 in n=3.
So, the ground state energy of this system is:
E = 2 ⋅ (E1 ) + 2 ⋅ (4E1 ) +1⋅ (9E1 ) = 19E1
„
What if we put 5 bosons in the well?
„
„
All 5 bosons can go into n=1!
So ground state energy is E=5E1
Example
„
For two electrons in an infinite potential well
well,
construct the total wavefunction of the ground
state.
state
„
F two electrons
For
l
in
i an infinite
i fi i potential
i l well,
ll
construct total wavefunctions of the first excited
state, with
i h and
d without
ih
consideration
id i off the
h
electron--electron Coulomb potential
electron
Summary/Announcements
„
Next time:
„
Multi--electrons
Multi
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HW 12 will be posted on Nov. 30th (Wednesday).
„
There will be a quiz next class (30th)