Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators? (ii) What is the general form of the hamiltonian operator? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators)? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various ‘historical’ formulae are shown to be particular instances of the general formulae proposed.Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators? (ii) What is the general form of the hamiltonian operator? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators)? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various ‘historical’ formulae are shown to be particular instances of the general formulae proposed.

Molecular PhysicsAug 20, 1985124 citations

The Operator Algebra and the Hamiltonian Formalism

Umang Mehta

Springer thesesJan 1, 20240 citations

Equations of Motion, Hamiltonian Operator and the Commutation Relations

WORLD SCIENTIFIC eBooksJan 1, 20110 citations

The non-Hermitian operators on the Baker-Hausdorff formula

Jack Whongius

This paper provides a connection to the non-Hermitian operators associated with the geometric potential function $s$ and Baker-Hausdorff formula. The geometric quantum potential is considered in a precise condition. The Ri-operator as a non-Hermitian Hamiltonian to describe the generalized quantum harmonic oscillator can be re-expressed as a more compact quantum formula by using the Baker-Hausdorff formula, more deeply, we prove some results based on the application of this formula.This paper provides a connection to the non-Hermitian operators associated with the geometric potential function $s$ and Baker-Hausdorff formula. The geometric quantum potential is considered in a precise condition. The Ri-operator as a non-Hermitian Hamiltonian to describe the generalized quantum harmonic oscillator can be re-expressed as a more compact quantum formula by using the Baker-Hausdorff formula, more deeply, we prove some results based on the application of this formula.

arXiv:2302.10116Jan 29, 20230 citations

Hamiltonian Operator

Christopher Witte

Compendium of Quantum PhysicsJan 1, 20090 citations

Refined formulation of quantum analysis,q-derivative and exponential splitting

Masuo Suzuki

Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential d f ( A ) satisfying the Leibniz rule. This formulation with use of the inner derivation δ A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q -derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q -derivative is also given.Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential d f ( A ) satisfying the Leibniz rule. This formulation with use of the inner derivation δ A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q -derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q -derivative is also given.

Journal of Physics A Mathematical and GeneralApr 24, 20065 citations

This paper provides a connection to the non-Hermitian operators associated with the geometric potential function

s

and Baker-Hausdorff formula. The geometric quantum potential is considered in a precise condition. The Ri-operator as a non-Hermitian Hamiltonian to describe the generalized quantum harmonic oscillator can be re-expressed as a more compact quantum formula by using the Baker-Hausdorff formula, more deeply, we prove some results based on the application of this formula.