Heston
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It seems like there should be a discrete-time discrete-space analog to the Schrödinger 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 Schrödinger 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 Schrödinger equation?
When I try that with the Schrödinger 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 Schrödinger equation?