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Discrete Analog of Schrodinger Equation?

  1. Mar 27, 2009 #1
    It seems like there should be a discrete-time discrete-space analog to the Schrodinger equation. For example, you can apply the classic explicit finite difference method to the heat equation and get a simple binomial or trinomial tree relationship in a lattice.

    When I try that with the Schrodinger equation (with zero potential), I find that the value of the wave function phi is a linear combination of adjacent values at the previous step, with complex weights -r*I/2, 1-r*I, and -r*I/2, where r is a constant and I is the imaginary unit. But this scheme does not conserve probability. The sum of squared absolute values of psi is not equal to 1.

    I know the Hermitian nature of the H-operator in continuous-time guarantees conservation of probability. This is not working in my discrete attempts. Is there any discrete analog of the Schrodinger equation?
     
  2. jcsd
  3. Mar 28, 2009 #2

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    I don't know about discrete time, but I've seen a discrete space version (propagation on a lattice) which gives the Schrodinger equation in the limit of zero spacing
     
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