Time dependent wave function

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Time dependent wave function We will assume that
potential V is independent of time and purely
radial
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Look for a stationary solution, i.e look for a
solution by separation of variables
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Easy to see solution is of the form A
solution of this type is a stationary state
solution
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Easy to see solution is of the form
E is a constant that comes from the separation of
variables which we identify with the energy And
satisfies the eigenvalue equation
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Remarks

• The time idependent Schrödinger equation is a 2nd
  order partial differential equation,
• It is an eigenvalue equation

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• Is zero outside ?0
• Hence rhs 0

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Flow of probabilty current through boundary
Rate of change of probability density within
volume
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Interpretation
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• If there is a zero barrier the entire beam passes
  through
• If there is an infinite barrier none of the beam
  passes through it is all reflected
• If there is a non-zero finite barrier
• It is partially transmitted and partially
  reflected
• Particles from the beam can tunnel out

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Classical Mechanics
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Quantum Mechanics
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Examples

• a decay of nuclei
• http//hyperphysics.phy-astr.gsu.edu/hbase/nuclear
  /alptun.html
• Josephson junction
• The Josephson effect is the phenomenon of current
  flow across two weakly coupled superconductors,
  separated by a very thin insulating barrier. This
  arrangementtwo superconductors linked by a
  non-conducting barrieris known as a Josephson
  junction the current that crosses the barrier is
  the Josephson current. It has important
  applications in quantum-mechanical circuits, such
  as SQUIDs.

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tabish.freeshell.org/physics/nboh/

• Trapping of H in Nb
• Niobium (Nb) is a metal which is superconducting
  below a temperature of 9.2K. Nb has a
  body-centered-cubic (BCC) structure. Hydrogen
  (H), being a small atom, if introduced in a Nb
  sample, can sit in the interstitial sites between
  the Nb atoms. At very low temperatures H is seen
  to move freely through the Nb lattice of atoms.
  However, if there are certain impurity atoms like
  Oxygen (O) or Nitrogen (N), present in the Nb
  sample, they form trapping centers for the
  otherwise freely moving hydrogen atoms. A typical
  location of an Oxygen atom in the Nb lattic is
  shown by in red in the figure.

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• In the presence of Oxygen impurities, the H atom
  likes to stay around such impurity sites. It
  tries to sit at the site where its energy will be
  the lowest. It turns out that the two sites shown
  in the figure are the lowest energy sites for the
  H atom (shown in white). These sites are
  physically quite close, and also energetically
  identical for the H atom. The hydrogen atom near
  such sites "sees" a potential which is like a
  double-well. However, in a given sample of Nb,
  there are many such double-wells because there
  are many H atoms trapped around Oxygen
  impurities. The presence of other impurities
  affect the symmetry of the double-well, and it
  will in general be asymmetric, as shown in the
  figure below. It can stay in either of the two
  wells, but not in between. At high temperatures H
  atom has enough energy to jump the barrier
  between the two wells. At low temperatures it can
  stay in either of the two wells, but cannot jump
  from one to the other.

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• Hydrogen atom tunnels between the different
  sitesLeft to itself at a particular site, the H
  atom will execute a coherent clock like motion
  between the two sites

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An other example

• Scanning Tunneling Microscope

The scanning tunneling microscope (STM) is a type
of electron microscope that shows
three-dimensional images of a sample. In the STM,
the structure of a surface is studied using a
stylus that scans the surface at a fixed distance
from it
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An extremely fine conducting probe is held close
to the sample. Electrons tunnel between the
surface and the stylus, producing an electrical
signal. The stylus is extremely sharp, the tip
being formed by one single atom. It slowly scans
across the surface at a distance of only an
atom's diameter. The stylus is raised and lowered
in order to keep the signal constant and maintain
the distance. This enables it to follow even the
smallest details of the surface it is scanning.
Recording the vertical movement of the stylus
makes it possible to study the structure of the
surface atom by atom. A profile of the surface is
created, and from that a computer-generated
contour map of the surface is produced.
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http//nobelprize.org/educational_games/physics/mi
croscopes/scanning/index.html
Xenon on NickelThis image shows xenon on a
nickel surface. The image is actually a
combination of two images. Defects in the nickel
surface were used to position the two images
correctly above each other
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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