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MathGroup Archive 2004

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RE: Quantum Mechanics, Boundary Value Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49580] RE: [mg49525] Quantum Mechanics, Boundary Value Problem
  • From: "Dr A.H. Harker" <a.harker at ucl.ac.uk>
  • Date: Sat, 24 Jul 2004 03:47:10 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear Anthony,

            I think you may be trying to be too sophisticated: from your
posting it seems as if you are remembering all that good stuff from the
first quantum mechanics course about continuity of the wavefunction and
its derivative and building that in explicitly at the boundary. Once
you've taken the plunge and decided to tackle the problem numerically,
there's no need for that -- you can just blast right across the whole
range with NDSolve:
se = -D[u[x], {x, 2}]/2 + If[x < 0, 0, 1]u[x] == .5 u[x]
NDSolve[{se, u[-10] == 1, u[10] == 0}, u[x], {x, -10, 10}]

 Note that this has arbitrarily set u to 1 at some point in the region
where there are incident and reflected running waves. The odds against a
point arbitrarily picked coinciding with a node of the wave pattern on
that side are pretty high, but if the point is close to a node you'll
end up with a wavefunction with quite a large amplitude.

 Of course, in your problem with the image potential there may also be
bound states at the surface, and there you would be into a different
game as you could not put u[x]==0 at both ends and expect to get
anything other than u[x] zero everywhere. 

   Tony Harker

 Dr A.H. Harker
 Director of Postgraduate Studies
 Deputy Head, Condensed Matter and Materials Physics Group
 Department of Physics and Astronomy
 University College London
 Gower Street
 LONDON
 WC1E  6BT
 (44)(0)207 679 3404
 a.harker at ucl.ac.uk


]->-----Original Message-----
]->From: anthony danese [mailto:danese at physics.rutgers.edu]
To: mathgroup at smc.vnet.net
]->Sent: 22 July 2004 07:47
]->To: mathgroup at smc.vnet.net
]->Subject: [mg49580] [mg49525] Quantum Mechanics, Boundary Value Problem
]->
]->Hi,
]->
]->I'm trying to find a numerical solution to Schrodinger's equation for
]->a nearly free electron potential (V(x)=Cos(x)) which meets an image
]->potential which is of the form (V(x)=C - 1/x) where C is a constant.
]->I can set up the problem using NDSolve, with 2 of my boundary
]->conditions being continuity of the wave function (psi) and its
]->derivative(psi') where the 2 potentials meet.  The problem is that I
]->need 2 other boundary conditions--I attempt to arbitrarily set psi = 0
]->far into the Cos(x)-like region of the potential and I would like psi
]->to vanish as for large values of x in the C-1/x region of space.  When
]->I try to run NDSolve, I either get the trivial psi = 0 or I get an
]->error telling me that the boundary conditions are not well defined.
]->
]->As a test, I have tried to run NDSolve to find psi for a step
]->potential where the step potential is greater than the energy of the
]->electron.  The solution should be free electron-like away from the
]->step and the solution should decay into the step.  This is a problem
]->that is easily solved analytically but Mathematica doesn't like the
]->way I'm stating the boundary conditions.  Is there anyway to solve
]->these particular examples of Schrodinger's equation with NDSolve?
]->
]->
]->Any help would be greatly appreciated,
]->
]->anthony danese



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