Some More Mathematics

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Some More Mathematics

• The Dirac Delta Function

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• It is not possible to develop a rigorous
  mathematical theory of integration where such a
  function exists i.e where the value of a function
  at one isolated point can affect the integral

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• The best way to think of the delta function is
  that it is an integral waiting to happen!

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Corollary
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Fourier INTegral Theorem
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Inverse Fourier Transform
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Convolution theorem
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Linear Operators, Wave Packets

• Landshoff Chapter 4

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Average value
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eigenvalue
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Proof can be extended to degenerate eigenvalues
if we chose to orthogonalize We have not assumed
B to be self adjoint If it is an observable then
its eigenvectors also form a complete set
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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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• 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