A quantum mechanical curvature theorem

Daniele Bonalda, Luigi Seveso, Matteo G. A. Paris

Jeremiah N. Silverman

Abstract The generalized Hellmann–Feynman and curvature theorems are derived for the generalized Sturm–Liouville‐type eigenvalue equation which is widely encountered in many branches of physics. In classical mechanics, the Sturm–Liouville eigenvalue equation primarily arises from applying the theory of small oscillations to vibration problems while, in quantum mechanics, the most important example is the time‐independent Schrödinger equation. The generalized theorems, respectively, provide simple useful expressions for the first and second λ derivatives of the eigenvalue where λ is an arbitrary real parameter appearing in the eigenvalue equation. These results, which apply to both classical and quantal systems, include as special cases the well‐known quantum mechanical Hellmann–Feynman and curvature theorems for the time‐independent Schrödinger equation, thus unifying the formalism. This connection between classical and quantum mechanics is of interest per se, and also because classical eigenvalue equations find application in the treatment of some quantum chemical problems. Two derivations of both generalizations are presented, the first proof applying to exact and Rayleigh–Ritz (linear variational) solutions of the eigenvalue problem, and the second, more general one, to arbitrary optimal variational as well as exact solutions. Necessary and sufficient conditions for the validity of the results are discussed. To demonstrate the insight they afford, several applications of the theorems in conjunction with perturbation theory are described.

Arturo Tozzi

The unexploited unification of general relativity and quantum mechanics (QM) prevents the proper understanding of the micro- and macroscopic world. Here we put forward a mathematical approach that introduces the problem in terms of negative curvature manifolds. We suggest that the oscillatory dynamics described by wave functions might take place on hyperbolic continuous manifolds, standing for the counterpart of QM’s Hilbert spaces. We describe how the tenets of QM, such as the observable A, the autostates ψa and the Schrodinger equation for the temporal evolution of states, might work very well on a Poincaré disk equipped with rotational groups. This curvature-based approach to QM, combined with the noncommutativity formulated in the language of gyrovectors, leads to a mathematical framework that might be useful in the investigation of relativity/QM relationships. Furthermore, we introduce a topological theorem, termed the punctured balloon theorem (PBT), which states that an orientable genus-1 surface cannot encompass disjoint points. PBT suggests that hyperbolic QM manifolds must be of genus ≥ 1 before measuring and genus zero after measuring. We discuss the implications of PBT in gauge theories and in the physics of the black holes.

E. R. Caianiello

A. R. Marlow

Arturo Tozzi

The unexploited unification of general relativity and quantum mechanics (QM) prevents the proper understanding of the micro- and macroscopic world. Here we put forward a mathematical approach that introduces the problem in terms of negative curvature manifolds. We suggest that the oscillatory dynamics described by wave functions might take place on hyperbolic continuous manifolds, standing for the counterpart of QM’s Hilbert spaces. We describe how the tenets of QM, such as the observable A, the autostates ψa and the Schrodinger equation for the temporal evolution of states, might work very well on a Poincaré disk equipped with rotational groups. This curvature-based approach to QM, combined with the noncommutativity formulated in the language of gyrovectors, leads to a mathematical framework that might be useful in the investigation of relativity/QM relationships. Furthermore, we introduce a topological theorem, termed the punctured balloon theorem (PBT), which states that an orientable genus-1 surface cannot encompass disjoint points. PBT suggests that hyperbolic QM manifolds must be of genus ≥ 1 before measuring and genus zero after measuring. We discuss the implications of PBT in gauge theories and in the physics of the black holes.