Symbolic-Manipulation Constructions of Hilbert-Space Metrics in Quantum Mechanics
نویسنده
چکیده
The problem of the determination of the Hilbert-space metric Θ which renders a given Hamiltonian H self-adjoint is addressed from the point of view of applicability of computer-assisted algebraic manipulations. An exactly solvable example of the so called Gegenbauerian quantum-lattice oscillator is recalled for the purpose. Both the construction of suitable Θ = Θ(H) (basically, the solution of the Dieudonne’s operator equation) and the determination of its domain of positivity are shown facilitated by the symbolic algebraic manipulations and by MAPLE-supported numerics and graphics.
منابع مشابه
کوانتش گرانش و بررسی هندسی مکانیک کوانتمی
We elaborate on some recent results on a solution of the Hilbert-space problem in minisuperspace quantum cosmology and discuss the consequences of making the (geometry of the) Hilbert space of ordinary nonrelativistic quantum systems time-dependent. The latter reveals a remarkable similarity between Quantum Mechanics and General Relativity.
متن کاملar X iv : 0 80 9 . 44 66 v 1 [ qu an t - ph ] 2 5 Se p 20 08 A term - rewriting system for computer quantum algebra
Existing computer algebra packages do not fully support quantum mechanics calculations in Dirac's notation. I present the foundation for building such support: a mathematical system for the symbolic manipulation of expressions used in the invariant formalism of quantum mechanics. I first describe the essential mathematical features of the Hilbert-space invariant formalism. This is followed by a...
متن کاملTwo sample constructions of all the Hilbert spaces for a given operator H of an observable
In Quantum Theory an operator H represents an observable provided only that it is self-adjoint in a Hilbert space L Θ equipped with a metric Θ. This means that for a given H , equation HΘ = ΘH specifies a complete menu of all the eligible Θ = Θ(H) needed to determine the inner product. We illustrate the feasibility of the construction of all of these Θs. It is based on the computerized symbolic...
متن کاملApplications of Finsler Geometry to Speed Limits to Quantum Information Processing
We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Here, we discuss speed limits in the context of quantum computing. We review some relevant parts of the theory of Finsler metrics on Lie groups and homogeneous spaces such as the spe...
متن کاملHamilton Operators, Discrete Symmetries, Brute Force and SymbolicC++
Abstract To find the discrete symmetries of a Hamilton operator Ĥ is of central importance in quantum theory. Here we describe and implement a brute force method to determine the discrete symmetries given by permutation matrices for Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi systems are considered as examples. A computer algebra implementation in SymbolicC++...
متن کامل