Theory in Materials Science

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Theory in Materials Science

• 3rd lecture
• Quantum Mechanics of Electrons

Associate Professor Koichi Kusakabe
Graduate School of Engineering Science, Osaka
University
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Purpose of todays lecture

• Electron hopping

The tight binding model
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What is the Quantum state ?

• We have a notion of a quantum state.
• We have an abstract notation of the state.

The name is 2px.
A wave function of the px state is
The space coordinate is r, q, j.
The spin coordinate is x.
Anyway, we know its existence!
This is another name of 2px.
This equation tells us how to make it!
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The quantum state

• Once we know existence of a quantum state, we can
  consider creation of the state from the vacuum!

is the vacuum.
This is a creation operator.
We create the 2px state from the vacuum.
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Wave functions in a carbon atom
Nodal surface x0.
No node along r.
2px.
r
Node at rrc.
2s.
r
en
This is a mean-field description!
A carbon atom
2px.
If we have six electrons in a carbon atom, each
of them should be in an eigen mode, i.e., an
eigen state in an independent particle
approximation.
2s.
1s.
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Number representation
en
A carbon atom
Since we have many quantum states as eigen
states, we have a notion using occupation with
a countable number of electrons in a state.
2px.
2s.
1s.
2px.
2px.
1s.
2s.
1s.
2s.
Up electrons
Down electrons
Furthermore,
We have another coordinate along the energy axis!
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A wave function as projection on another state

• A wave function is described by two coordinates.
• This understanding implies existence of the
  next interpretation, a wave function as a
  projection.

Furthermore,
A field operator
The wave function is a probability amplitude to
find an electron in a state 2px? at a coordinate
(r,q,j,x).
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Many electron systems in materials
3su
C6H6
1pg
2p
3sg
2su
r
2s
1pu
2sg
1su
C2 a molecule
1s- anti-bonding
1s
1s-bonding
1sg
r
In a benzene molecule, an electron state is
created from 2pz orbitals. (The above w.f. is an
anti-bonding w.f.)
If we have two carbon atoms, the eigen states are
remade in a linear combination of atomic orbitals.
We need a compact description of many electron
systems.
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Many-body wavefunction
A wave function of two particles in one dimension
The left configuration
x2
x2
x2x1 inflection points
x1
x
In a classical mechanics, we have two coordinates
for two particles.
x1
In quantum mechanics, we have a wave function of
two particles in two dimensional space.

• N particles in d dimensions
• ? w.f. in N d dimensional space.
• The Pauli principles have to be satisfied.
• ? Anti-symmetry for electrons

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.

Expand each xi dependence by jl(xi).
This equation tells us that a general coefficient
is given by a coefficient times a sign of a
permutation.
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The Slater determinant
The sign of a permutation P.
The Slater determinant
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The number representation
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Particle exchange in a Slater determinant.
This anti-symmetric wave function has a next
character.
Interchanging two coordinates results in a sign
change!
Are there any compact number representation
showing the sign change, when two coordinates are
interchanged? ? Yes!
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Introduction to the 2nd quantization I.
n
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Introduction to the 2nd quantization II.
0
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Introduction to the 2nd quantization III.
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Introduction to the 2nd quantization IV.
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Introduction to the 2nd quantization V.
k
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Introduction to the 2nd quantization VI.
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Introduction to the 2nd quantization VII.
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Bloch solution for an effective model
Using a proper single-particle description of the
electronic structure, we have Bloch waves as a
solution. Cf. DFT, HF.
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The Wannier state A unitary transformation
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Two sets of operators Bloch v.s. Wannier
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The canonical anti-commutation relation
We now introduce a second-quantization
representation of an effective Hamiltonian.
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mks
mks
mks
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The tight-binding model
?
?
t
?
?
?
?
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Schematic pictures of the 2nd quantization
Wave functions (1st quantization)
Motion of a particle
Description in classical mechanics
Description by 2nd quantization
Abstract vector (2nd quantization)
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Creation of an electron at a position
A quantum mechanical interaction process is
described by a diagram.
interaction
A propagating wave
These creation and annihilation operators are
used in the field theory.