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Ben Gurion University of the Negev
www.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter
Physics 3 for Electrical Engineering
Lecturers: Daniel Rohrlich, Ron Folman
Teaching Assistants: Daniel Ariad, Barukh Dolgin
Week 11. Quantum mechanics – Schrödinger’s equation for multiple
electrons • identical particles and the Pauli principle • atomic ground
states • periodic table
Sources: Feynman Lectures III, Chap. 19 Sect. 6;
3 ‫ פרק‬,8 ‫ יחידה‬,‫פרקים בפיסיקה מודרנית‬
Tipler and Llewellyn, Chap. 7 Sects. 6-8.
A few quantum principles
1. The classical distinction between particles and waves breaks
down [“wave-particle duality”].
A few quantum principles
1. The classical distinction between particles and waves breaks
down [“wave-particle duality”].
2. Physical states are normalized vectors ψ(r), Ψ(r,t), ψ , (t)
[ → superposition principle].
A few quantum principles
1. The classical distinction between particles and waves breaks
down [“wave-particle duality”].
2. Physical states are normalized vectors ψ(r), Ψ(r,t), ψ , (t)
[ → superposition principle].
3. Measurable physical quantities – “observables” – correspond
to Hermitian or (self-adjoint) operators on the state vectors.
A few quantum principles
1. The classical distinction between particles and waves breaks
down [“wave-particle duality”].
2. Physical states are normalized vectors ψ(r), Ψ(r,t), ψ , (t)
[ → superposition principle].
3. Measurable physical quantities – “observables” – correspond
to Hermitian or (self-adjoint) operators on the state vectors.
4. If a system is an eigenstate a with eigenvalue a of an
observable  , then a measurement of  on a will yield a.
A few quantum principles
1. The classical distinction between particles and waves breaks
down [“wave-particle duality”].
2. Physical states are normalized vectors ψ(r), Ψ(r,t), ψ , (t)
[ → superposition principle].
3. Measurable physical quantities – “observables” – correspond
to Hermitian or (self-adjoint) operators on the state vectors.
4. If a system is an eigenstate a with eigenvalue a of an
observable  , then a measurement of  on a will yield a.
Conversely, if a measurement of  on any state yields a, the
measurement leaves the system in an eigenstate a .
A few quantum principles
5. The probability that a system in a normalized state ψ can be
found in the state φ is φ ψ
2
[Born probability rule].
6. The time evolution of a quantum state (t ) is given by

i
 (t )  Hˆ  (t )
t
,
where Ĥ is the Hamiltonian (kinetic energy + potential energy)
of the system in the state (t ) [Schrödinger’s equation].
Schrödinger’s equation for helium
The Schrödinger equation for helium is
 2 2
2e 2 
E (r1 , r2 )    1 
  (r1 , r2 )
4 0 r1 
 2
 2 2
2e 2 
  2 
  (r1 , r2 )
4 0 r2 
 2
e2

 (r1 , r2 ) .
4 0 r1  r2
Schrödinger’s equation for helium
The Schrödinger equation for helium is
 2 2
2e 2 
E (r1 , r2 )    1 
  (r1 , r2 )
4 0 r1 
 2
 2 2
2e 2 
  2 
  (r1 , r2 )
4 0 r2 
 2
e2

 (r1 , r2 ) .
4 0 r1  r2
Schrödinger’s equation for helium
As usual, we’d like to write Ψ(r1,r2) as a product ψ1(r1)ψ2(r2),
 2 2
2e 2 
E (r1 , r2 )    1 
  (r1 , r2 )
4 0 r1 
 2
 2 2
2e 2 
  2 
  (r1 , r2 )
4 0 r2 
 2
e2

 (r1 , r2 ) ,
4 0 r1  r2
but the electron-electron repulsion gets in the way.
Schrödinger’s equation for helium
We can write Ψ(r1,r2) = ψ1(r1)ψ2(r2) anyway,
 2 2
2e 2 
E (r1 , r2 )    1 
  (r1 , r2 )
4 0 r1 
 2
 2 2
2e 2 
  2 
  (r1 , r2 )
4 0 r2 
 2
e2

 (r1 , r2 ) ,
4 0 r1  r2
and treat the electron-electron repulsion as a “perturbation”.
Schrödinger’s equation for helium
We can write Ψ(r1,r2) = ψ1(r1)ψ2(r2) anyway,
 2 2
2e 2 
Eψ1 (r1 )ψ 2 (r2 )    1 
 ψ1 (r1 )ψ 2 (r2 )
4 0 r1 
 2 
 2 2
2e 2 
  2 
 ψ1 (r1 )ψ 2 (r2 )
4 0 r2 
 2 
e2

ψ1 (r1 )ψ 2 (r2 ) ,
4 0 r1  r2
and treat the electron-electron repulsion as a “perturbation”.
Schrödinger’s equation for helium
Let ψ1 = ψ
Eψ
n1l1m1
ψ
and ψ2 = ψ
n1l1m1 n2l2 m2
n2l2m2
. Then
 13 .6 eV 13 .6 eV 
 4  

ψ n l m ψ n l m
2
2
n1
n2  1 1 1 2 2 2

e2

ψ
ψ
n
l
m
1 1 1 n2l2 m2
4 0 r1  r2
,
and we can take the scalar product of both sides of this equation
with ψ
ψ
n1l1m1 n2l2m2
to estimate E as an expectation value:
Schrödinger’s equation for helium
E
54.4 eV
n12

54 .4 eV
n22
e2

4 0 r1  r2
.
The expectation value is, explicitly,

ψ*
ψ*
n1l1m1 n2l2 m2
e2
ψ
ψ
d 3 x1d 3 x2 .
4 0 r1  r2 n1l1m1 n2l2m2
Schrödinger’s equation for multiple electrons
The Schrödinger equation for an atom with Z protons is
Z  2
2

Ze
E (r1 , r2 ,..., rZ )    
 2j 
  2
4 0 r j
j

1



  (r , r ,..., r )
1 2
Z


 k 1 Z

2
e

  (r1 , r2 ,..., rZ ) .


4 0 r j  rk 
j

1
k

1



Schrödinger’s equation for multiple electrons
The Schrödinger equation for an atom with Z protons is
Z  2
2

Ze
E (r1 , r2 ,..., rZ )    
 2j 
  2
4 0 r j
j

1



  (r , r ,..., r )
1 2
Z




2
e

  (r1 , r2 ,..., rZ ) .


4 0 r j  rk 
j

k



Identical particles and the Pauli principle
A new quantum principle – hard to understand but easy to use –
applies only to identical particles.
(We did not need this principle for the hydrogen atom, which
consists of one proton and one electron, because these particles
are not identical.)
Identical particles and the Pauli principle
A new quantum principle – hard to understand but easy to use –
applies only to identical particles.
(We did not need this principle for the hydrogen atom, which
consists of one proton and one electron, because these particles
are not identical.)
Quantum states must be symmetric under exchange of any two
identical bosons and antisymmetric under exchange of any two
identical fermions.
Bosons: photons, pions, ….
Fermions: electrons, protons, neutrons,….
Identical particles and the Pauli principle
A new quantum principle – hard to understand but easy to use –
applies only to identical particles.
(We did not need this principle for the hydrogen atom, which
consists of one proton and one electron, because these particles
are not identical.)
Quantum states must be symmetric under exchange of any two
identical bosons and antisymmetric under exchange of any two
identical fermions.
Bosons: photons, pions, …(spin 0, 1,...).
Fermions: electrons, protons, neutrons,…(spin 1 2,
3
2
,… ).
Identical particles and the Pauli principle
Intuitive connection between spin and symmetry: Consider two
ends of a ribbon; exchange the two ends, without rotating either
end.
Identical particles and the Pauli principle
Intuitive connection between spin and symmetry: Consider two
ends of a ribbon; exchange the two ends, without rotating either
end.
There is now a 2π twist in the ribbon!
Identical particles and the Pauli principle
If the “ends of the ribbon” have spin 1 2, 3 2 ,… then a 2π rotation
yields a – sign because ei2π(1/2) = ei2π(3/2) =…= –1:
→ antisymmetric wave function.
If the “ends of the ribbon” have spin 0, 1,… then a 2π rotation
yields a + sign because ei2π(0) = ei2π(1) =…= 1:
→ symmetric wave function.
Identical particles and the Pauli principle
An infinite square well contains two particles. What are the
possible states of the particles? Neglect their interactions.
The single-particle states are ψ n ( x) 
outside
inside
0
2
nx
sin
.
L
L
outside
L
x
Identical particles and the Pauli principle
Possible states of an electron and a neutron: products of states
ψ j ( x e )ψk ( x n ) for any j and k, and any combination of these
products.
outside
inside
0
outside
L
x
Identical particles and the Pauli principle
Possible states of two neutrons: antisymmetrized products of
1
states
ψ j ( x1)ψ k ( x2 )  ψ k ( x1)ψ j ( x2 ) for any j ≠ k, and
2
any combination of these antisymmetrized products.


outside
inside
0
outside
L
x
Identical particles and the Pauli principle
Possible states of two photons: symmetrized products of
1
states
ψ j ( x1)ψ k ( x2 )  ψ k ( x1)ψ j ( x2 ) for any j and k,
2
and any combination of these symmetrized products.


outside
inside
0
outside
L
x
Identical particles and the Pauli principle
Can two neutrons be in the same state?
outside
inside
0
outside
L
x
Identical particles and the Pauli principle
Can two neutrons be in the same state ψj(x)?
1
No, because
ψ j ( x1)ψ j ( x2 )  ψ j ( x1)ψ j ( x2 )  0.
2


outside
inside
0
outside
L
x
Identical particles and the Pauli principle
Can two neutrons be in the same state ψj(x)?
1
No, because
ψ j ( x1)ψ j ( x2 )  ψ j ( x1)ψ j ( x2 )  0.
2


Pauli’s exclusion principle: no two fermions in the same state.
Identical particles and the Pauli principle
Can two neutrons be in the same state ψj(x)?
1
No, because
ψ j ( x1)ψ j ( x2 )  ψ j ( x1)ψ j ( x2 )  0.
2


Pauli’s exclusion principle: no two fermions in the same state.
“Exchange repulsion”: What is the relative probability that two
1
fermions in the state
ψ j ( x1)ψ k ( x2 )  ψ k ( x1)ψ j ( x2 )
2
will be found at the same point x?


Identical particles and the Pauli principle
Can two neutrons be in the same state ψj(x)?
1
No, because
ψ j ( x1)ψ j ( x2 )  ψ j ( x1)ψ j ( x2 )  0.
2


Pauli’s exclusion principle: no two fermions in the same state.
“Exchange repulsion”: What is the relative probability that two
1
fermions in the state
ψ j ( x1)ψ k ( x2 )  ψ k ( x1)ψ j ( x2 )
2
will be found at the same point x? It is

2
1
ψ j ( x)ψ k ( x)  ψ k ( x)ψ j ( x)  0 .
2

Atomic ground states
The ground state of an atom is the lowest-energy state. What is
the ground state of helium (Z = 2)?
Atomic ground states
The ground state of an atom is the lowest-energy state. What is
the ground state of helium (Z = 2)?
It is approximately
1
2
ψ1,0,0,1/ 2 ( x1)ψ1,0,0,1/ 2 ( x2 )  ψ1,0,0,1/ 2 ( x1)ψ1,0,0,1/ 2 ( x2 )
which we can write as

ψ
2
1
or even
100,  1


2
1
1

2
ψ100, 
 
1

2

2
ψ100,  1 ψ100, 
ψ
100 1
ψ100
2
2
.

Atomic ground states
The ground state of an atom is the lowest-energy state. What is
the ground state of helium (Z = 2)?
It is approximately
1
2
ψ1,0,0,1/ 2 ( x1)ψ1,0,0,1/ 2 ( x2 )  ψ1,0,0,1/ 2 ( x1)ψ1,0,0,1/ 2 ( x2 )
which we can write as
1
    
ψ
ψ
.
100
100
1
2
1
2
1
2
2
The spin state is a singlet and corresponds to j = 0. (Show that


both Jˆ  Sˆ(1)  Sˆ(2) and Jˆ  Sˆ(1)  Sˆ(2) annihilate this state.)
Atomic ground states
Exercise: What is the ground state energy of helium, neglecting
the electron-electron interaction?
Atomic ground states
Exercise: What is the ground state energy of helium, neglecting
the electron-electron interaction?
Solution: Neglecting the electron-electron interaction, each
electron has energy
En  
 Z 2e 4
4 0 
2
2 2
2 n

2 13 .6 eV
Z
2
,
n
with n = 1 and Z = 2. Hence the ground-state energy is
8 × (–13.6 eV) = –108.8 eV.
The experimentally measured value of the ground-state energy
is –78.95 eV. (We expect it to be higher than our solution
because the electron-electron interaction is repulsive.)
Periodic table
The ground states of atoms are obtained by filling up the states,
one electron to a state, in order of their energy. Although the
ground states and their energies are perturbed from the singleelectron states and energies that we labeled according to the
quantum numbers n,l,m,ms, we still use these quantum numbers
to label the perturbed states. A shell contains all states with the
same principal quantum number n; an orbital contains all states
with the same n and l. A superscript indicates the population of
each orbital.
How many electrons can an orbital hold?
How many electrons can a shell hold?
Periodic table
The ground states of atoms are obtained by filling up the states,
one electron to a state, in order of their energy. Although the
ground states and their energies are perturbed from the singleelectron states and energies that we labeled according to the
quantum numbers n,l,m,ms, we still use these quantum numbers
to label the perturbed states. A shell contains all states with the
same principal quantum number n; an orbital contains all states
with the same n and l. A superscript indicates the population of
each orbital.
How many electrons can an orbital hold? 2(2l+1)
How many electrons can a shell hold? 2n2
Periodic table
The ground states of atoms are obtained by filling up the states,
one electron to a state, in order of their energy. Although the
ground states and their energies are perturbed from the singleelectron states and energies that we labeled according to the
quantum numbers n,l,m,ms, we still use these quantum numbers
to label the perturbed states. A shell contains all states with the
same principal quantum number n; an orbital contains all states
with the same n and l. A superscript indicates the population of
each orbital.
Examples:
H – 1s1
Li – 1s22s1
Na – 1s22s22p63s1
He – 1s2
Ne – 1s22s22p6
Cl – 1s22s22p63s23p5
Periodic table
The spin-orbit coupling breaks the degeneracy of the orbitals
within a shell. As l increases within a shell, so does the energy.
The chemical properties of an atom depend largely on the
valence electrons, which populate the highest-energy states and
are least attached to the atom.
Compare:
F – 1s22s22p5
Cl – 1s22s22p63s23p5
Ne – 1s22s22p6
Ar – 1s22s22p63s23p6
Na – 1s22s22p63s1
K – 1s22s22p63s23p64s1
Periodic table
Energy
Some shells overlap at small Z.
More about identical particles and the Pauli principle:
Consider an interferometer with four half-silvered mirrors.
ψ
φ
1
2
Two particles enter from opposite corners. Assume, at first, that
they are not identical. Each reflection induces a phase factor i.
Initial state (if the particles are not identical):
ψ1φ 2
ψ
φ
2
1
Final state (if the particles are not identical):
1
L 1 i U 1 i D 1  R 1 i U 2  L 2  R 2  i D 2
4
U
ψ1


L
φ
R
2
D
(Assume that the path lengths are all equal.)

Final state – measuring correlations
U
〰
φ
ψ
1
L
R
〰
2
D
〰
Final state (if the particles are not identical):
1
L 1 i U 1 i D 1  R 1 i U 2  L 2  R 2  i D 2
4
But if the particles are identical, we must either symmetrize this
state (e.g. for photons) or antisymmetrize it (e.g. for electrons).


Symmetrized state:
1
L 1 i U 1 i D 1  R
4 2


1
4 2

1
2 2
1
 i U
2
 L
2
 R
 i U 1  L 1  R 1  i D 1  L 2  i U
[ L 1 L
 L
1

R
2
2
U
 R
1
1
L
U
2
2
 D
U
1
1
D
D
2
2
 R
1
 D 1U
R
2
]
2
2
 iD
2
i D
2
2

 R
2

Final state (if the particles are not identical):
1
L 1 i U 1 i D 1  R 1 i U 2  L 2  R 2  i D 2
4
But if the particles are identical, we must either symmetrize this
state (e.g. for photons) or antisymmetrize it (e.g. for electrons).



Symmetrized state:
Pairs of photons exit the interferometer in the same state (same
port) at the same corner, or in opposite directions (L/R or U/D)
at ports on opposite corners.
Final state (if the particles are not identical):
1
L 1 i U 1 i D 1  R 1 i U 2  L 2  R 2  i D 2
4
But if the particles are identical, we must either symmetrize this
state (e.g. for photons) or antisymmetrize it (e.g. for electrons).


Antisymmetrized state:
1
L 1 i U 1 i D 1  R
4 2


1
4 2

1
2 2
1
 i U
2
 L
2
 R
 i U 1  L 1  R 1  i D 1  L 2  i U
[i L 1 U
i L
1

D
2
2
i U
i D
1
1
L
L
2
2
i D
i U
1
1
R
R
2
2
i R
1
D
i R 1 U
2
2
 iD
2
2
]
i D
2
2

 R
2

Final state (if the particles are not identical):
1
L 1 i U 1 i D 1  R 1 i U 2  L 2  R 2  i D 2
4
But if the particles are identical, we must either symmetrize this
state (e.g. for photons) or antisymmetrize it (e.g. for electrons).



Antisymmetrized state:
Pairs of electrons exit the interferometer in different states
(different ports) at the same corner, or in orthogonal directions
(L/D or U/R) at ports on opposite corners.