Re: Schrodinger Equation Reversibility and Additivity

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 01/08/05 

Date: Sat, 8 Jan 2005 14:42:59 +0000 (UTC)

On 7 Jan 05 15:53:48 -0500 (EST), Osher Doctorow wrote:
>In my recent thread on the Schrodinger Equation, I outlined a
>derivation of the equation by dimensional analysis with and
>without a probability dimension for the wave function.
>Miguel A. Martin-Delgado in "The Schrodinger equation, reversibil-
>ity, and the Grover algorithm," arXiv:quant-ph/0412130 v1 16
>Dec 2004 (he's at U. Complutense, Madrid) has independently shown
>how the Schrodinger equation is importantly related to reversibil-
>ity and additivity-subtractivity rather than multiplication-
>division. It is of course desirable to use addition and subtrac-

Martin-Delgado points out that quantum computation is a class of
reversible computation and that quantum programming harnesses this
reversibility, and that entanglement results from combining QM
(Quantum Mechanical) Supersition Principle with single-particle
tensor products to store many classical registers simultaneously
in quantum paarallelism. Unlike some other researchers, Martin-
Delgado points out that QM is probabilistic and this implies that
Quantum Computation is also and that the object of quantum prog-
ramming is to make this probability as close to 1 as possible
with a pattern of constructive interference of amplitudes as in
the Shor (factoring) algorithm and the Grover (item search in
disordered database) algorithm.

With reversibility, we can compute backwards exactly by saving
only the number of steps and the final conditions, unlike current
standard irreversible computers. Quantum computers (QC) operate
with finiteness by putting the Schrodinger equation on a space-
time lattice which is discrete. There are two types of finite
difference schemes which work, the asymmetric difference scheme
and the symmetric difference scheme, the former leading to
approximate but not exact reversibility, the latter to exact
reversibility, with respective approximations to derivatives:

1) (d/dz)F(z) = [F(z_i+1) - F(z_i)]/e + 0(e)
2) (d/dz)F(z) = (1/(2e))[F(z_i+1) - F(z_i-1)] + 0(e^2)

for a given function F(z).

Osher Doctorow

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