Here is our first attempt to model the abstract Dirac's
formalism of Quantum Mechanics in Haskell.
www.numeric-quest.com/haskell/QuantumVector.html
The exerpt from the summary follows.
Jan Skibinski
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We recognize a quantum state as an abstract vector | x >,
which can be represented in one of many possible bases --
similar to many alternative representations of a 3D vector
in rotated systems of coordinates. A choice of a particular
basis is controlled by a generic type variable, which can
be any Haskell object -- providing that it supports a notion
of equality and ordering.
The base vectors are abstract: on one hand they are just
used for identification purposes, on another -- they obey
all the rules of a vector space. Any vector | x > can
be represented as a linear combination of the base vectors
and complex scalars. [..]
We only require and impose the condition, that any two
base vectors from the same basis are orthonormal, as in:
< (i, j) | (p, q) > = d (i, j) (p, q)
where the left hand side is a scalar product and on the
right is a generalized definition of the classical Kronecker's
delta.
With this abstract notion we proceed with Haskell definition
of two vector spaces: Ket and its dual Bra. We demonstrate
that both are properly defined according to the abstract
mathematical definition of vector spaces. We then introduce
inner product and demonstrate that our Bra and Ket can be
indeed considered the vector spaces with inner product.
Multitude of examples are attached in the description. To
verify the abstract machinery developed here we also
provide the basic library module Momenta -- a non-trivial
example designed to compute Clebsch-Gordan coefficients
of a transformation from one basis of angular momenta to
another.
