Schrdinger Equation

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Schrödinger Equation
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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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Proof
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• Discrete energy levels as in Bohr atom

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Propability interpretation
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Run into a problem
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• Not normalizable if we are talking about a
  particle in free space
• Can deal with it if we normalize in a finite size
  box

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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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Particle in a box
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Boundary Conditions
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b
U
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In the diagram we show the intersection of the
curves -12mU/?2h2 with tan2?b/2 and
cotan2?b/2 The curves only intersect at a finite
number of pointsgt Finite number of allowed
values for E The number of such points depends on
the magnitude of U and b Note if Ult0 then there
are no bound state solutions (U is repulsive, not
attractive)
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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.