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List:       haskell
Subject:    Module QuantumVector
From:       Jan Skibinski <jans () numeric-quest ! com>
Date:       2000-05-31 16:37:55
Message-ID: Pine.LNX.4.21.0005310734050.10139-100000 () info ! numeric-quest ! com
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	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

	-----------------------------------------------------------  

	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.
	





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