Average value

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Average value
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Canonical Commutator Relation x,pih
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eigenvalue
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Ehrenfests theorem
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Recall Hydrogen Atom
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• Electrons, protons, neutrons are all spin ½
  particles
• Fermions (such as the electron, proton,neutron)
  have half-integer values, whereas bosons (e.g.
  photon, mesons) have integer spin values.

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Pauli Exclusion Principle

• It states that no two identical fermions may
  occupy the same quantum state simultaneously. A
  more rigorous statement of this principle is
  that, for two identical fermions, the total wave
  function is anti-symmetric. For electrons in a
  single atom, it states that no two electrons can
  have the same four quantum numbers, that is, if
  n, l, and ml are the same, sz must be different
  such that the electrons have opposite spins.

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Block waves

• In a metal the atoms release their electrons
  which flow around the nuclei
• Which are at fixed lattice sites
• The thus move in a potential which is periodic
• The perodicity beging defined by the underlying
  symmetry of the solids crystaline structure

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This image of a lattice crystal was captured by
Cornell researchers using a scanning transmission
electron microscope (STEM) at IBM. The yellow
circles in the center of each pear-shaped
molecule represent the stronger signal produced
by a large atom the red portions that make up
the top of each pear shape show the weaker signal
of the smaller atoms. The image allows
researchers to see the orientation of the
individual atoms within a crystal for the first
time, thus giving researchers a vital tool for
predicting the crystal's properties. A model of
the molecular structure is superimposed on the
image.
http//www.news.cornell.edu/stories/June06/Mkhoyan
_polarity.lg.html
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• Consider a one dimensional crystal
• The periodicity condition is given by
• H(xL)H(x)
• Where H is the hamiltonian and L is a constant
• Define the translation operator
• Df(x)f(xL)

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• DH(x)?(x)H(xL)?(xL)H(x) ?(xL)
• HD ?(x)
• gtH,D0
• Let ?n be a stationary state
• H ?n En ?n

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a-L
-L
x
L
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Plot of f(E) for a typyical set of a,L, U
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A crystal consists of a collection of atoms
arranged in a regular array The spacing between
the atoms being of the same order as the
dimensions of the atoms To all intents and
purposes the atoms are fixed at these lattice
points An electron fixed to an atom has discrete
energies .Imagine that we can assemble a Set of
discrete atoms whose spacing.L, can be altered at
will If Lgtgt1 then the motion of an electron in
one atom will be unaffected by the Electrons in
the other atom but as L is decreased, as we will
see,each individual energy level spreads out into
a band of closely spaced levels. These bands are
separated by energy gaps that are forbidden to
the electrons
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Remarks

• The bands corresponding to the lowest energies
  are narrowest, more atomic like
• Which we interpret as the electrons nearest the
  nucleus are the least affected
• The band structure persists even for Egt0
• The energy gap decreases as E increases
• but they can still be of appreciable width
  even if the electron has nearly enough energy to
  escape

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E/U
Left allowed energies for an electron in a
single potential well Right Allowed energies for
an array of periodically spaced
wells 2mUL2/h2121, barrier thicknessL/16
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Any solid has a large number of bands. In
theory, it can be said to have infinitely many
bands (just as an atom has infinitely many energy
levels). However,
all but a few lie at energies so high that any
electron that reaches those energies escapes
from the solid. These bands are usually
disregarded. Bands have different widths, based
upon the properties of the atomic orbitals from
which they arise. Also, allowed bands may
overlap, producing (for practical purposes) a
single large band.
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• Metals contain a band that is partly empty and
  partly filled regardless of temperature.
  Therefore they have very high conductivity.
• The lowermost, almost fully occupied band in an
  insulator or semiconductor is called the valence
  band by analogy with the valence electrons of
  individual atoms. The uppermost, almost
  unoccupied band is called the conduction band
  because only when electrons are excited to the
  conduction band can current flow in these
  materials. The difference between insulators and
  semiconductors is only that the forbidden band
  gap between the valence band and conduction band
  is larger in an insulator, so that fewer
  electrons are found there and the electrical
  conductivity is lower. Because one of the main
  mechanisms for electrons to be excited to the
  conduction band is due to thermal energy, the
  conductivity of semiconductors is strongly
  dependent on the temperature of the material.

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Boundary conditions

• Even for the smallest it is found that the
  properties of the crystal and disposition of
  allowed energies do not depend critically on the
  boundary conditions at the surface
• The simplest choice of boundary condition is to
  require ?0 on the faces(or in our linear model
  at the end of the chain)

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Disadvantage

• Our Block waves will not satisfy this condition
• It is true we could take combinations of Block
  waves and achieve a standing wave but this makes
  the problem more complex
• And it is difficult to then describe the flow of
  energy or charge around the crystal,
• The travelling wave ?(x)eikxuk(x) is much better
  for this

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Periodic Boundary Conditions

• The wave function is required to take the same
  value at corresponding points on opposite faces
  of the crystalthe wave simultaneously enters and
  leaves the crystal
• For all but the smallest nano-crystals this has
  negligible effect on the band structure or
  energies

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In the linear model the periodic boundary
condition is equivalent to bending the chain of
atoms around to form a closed loop.Choosing the
coordinates of the chain st the end of the chains
are x0, xNL, We require ?(0)?(NL) gtexpikNL1
gt k2n?/NL, n0,?1,?2,(1)
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• Going back to our derivation of the dispersion
  relation we see that ? and the corresponding E
  are unchanged if k is either increased or
  decreased by 2?/L
• Thus we may, without losing any states confine k,
  to an interval of length 2?/L
• For example
• -?/Lltk? ?/L

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• As k takes the allowed values(1)that fall within
  this interval we pass through every one of the
  allowed levels in each band. Thus each band has
  just N allowed levels, where N is the total
  number of atoms in the crystal

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• When the atoms are very far apart each band
  consists of a single level the energy level of
  the solitary atom. However this single level
  corresponds to N different states, because each
  electron can be attached to any one of N
  different atoms.
• As the atoms are brought closer together the N
  fold degenerate levels spreads out into bands

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• In many cases the crystal lattice is not a
  regular spaced array of single atoms but rather
  an array of unit cells where cell consists of
  more than one atom.

Generally the number of levels in each band is
equal to the number of unit cells in the crystal
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Band Overlap We are interested in finding the
solution to the dispersion relation f(E)coskL i.e
we wish to be able to express E as a function of
k A given value of k corresponds to many values
of E In our 1-D example the number for which
Elt0 Corresponds to the depth and width of the
potential wells But for Egt0 there are an infinite
number. Since k only enters through the term
coskL The solutions for E are symmetric about the
origin in k

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In our 1D model the edges of the allowed bands
occur when Cos kL?1 and given our range of
definition -?/Lltk? ?/L This corresponds to k0 or
k ??/L
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Dispersion curves corresponding to
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In the one dimensional model, gaps always occur
between the bands , regardless of how high up in
the energy bands we are. However in 3 dimensions
the bands may overlap. The block wavefunction
becomes ?(r) expik.ruk(r) k is a vector. The
dispersion curves become 3 dimensional surfaces
in the 4 Dimensional (E,k) space The properties
of the crystal may be different in different
directions. If we draw sections of this surface
corresponding to different directions of k the
resulting curves need not be the same. Each such
section will have the same general appearance
with bands and gaps but the position of the gaps
and their widths may vary with the direction
chosen for k
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Any solid has a large number of bands. In
theory, it can be said to have infinitely many
bands (just as an atom has infinitely many energy
levels). However,
all but a few lie at energies so high that any
electron that reaches those energies escapes
from the solid. These bands are usually
disregarded. Bands have different widths, based
upon the properties of the atomic orbitals from
which they arise. Also, allowed bands may
overlap, producing (for practical purposes) a
single large band.
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• Metals contain a band that is partly empty and
  partly filled regardless of temperature.
  Therefore they have very high conductivity.
• The lowermost, almost fully occupied band in an
  insulator or semiconductor is called the valence
  band by analogy with the valence electrons of
  individual atoms. The uppermost, almost
  unoccupied band is called the conduction band
  because only when electrons are excited to the
  conduction band can current flow in these
  materials. The difference between insulators and
  semiconductors is only that the forbidden band
  gap between the valence band and conduction band
  is larger in an insulator, so that fewer
  electrons are found there and the electrical
  conductivity is lower. Because one of the main
  mechanisms for electrons to be excited to the
  conduction band is due to thermal energy, the
  conductivity of semiconductors is strongly
  dependent on the temperature of the material.
• The Fermi energy is a the energy of the highest
  occupied quantum state in a system of fermions at
  absolute zero temperature

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• http//hyperphysics.phy-astr.gsu.edu/hbase/solids/
  fermi.html