Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the exact discretization of Schr\"odinger equation   We show that the exact discrete analogue of Schr\"odinger equation can be
derived naturally from the Hamiltonian operator of a Schr\"odinger field theory
by using the discrete Fourier transform that transforms the operator from
momentum representation into position representation. The standard central
difference equation that is often used as the discretized Schr\"odinger
equation actually describes a different theory since it is derived from a
different Hamiltonian operator. The commutator relation between the position
and momentum operators in discrete space is also derived and found to be
different from the conventional commutator relation in continuous space. A
comparison between the two discretization formulas is made by numerically
studying the transmission probability for a wave packet passing through a
square potential barrier in one dimensional space. Both discretization formulas
are shown to give sensible and accurate numerical results as compared to
theoretical calculation, though it takes more computation time when using the
exact discretization formula. The average wave number $k_0$ of the incident
wave packet must satisfy $|k_0\ell| < 0.35$, where $\ell$ is the lattice
spacing in position space, in order to obtain an accurate numerical result by
using the standard central difference formula.

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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
1
The categories of this paper are: cs
16
The categories of this paper are: stat
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The categories of this paper are: nlin
7
The categories of this paper are: hep-th
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The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
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The categories of this paper are: math
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: econ