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arXiv:1605.06534 · IJEIR

Emergent non-Hermitian contributions to the Ehrenfest and Hellmann-Feynman theorems

Georgios S. Konstantinou, Kyriakos Kyriakou, Konstantinos Moulopoulos

Published May 17, 2016• 5 citations
arXivHTMLPDF

Abstract

We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in an arbitrary parameter-space, in the sense of Berry's adiabatic and cyclic procedures). In this short note, we report modifications of these two theorems when such non-Hermiticities appear, and we demonstrate how they resolve certain Quantum Mechanical paradoxes (most of them having been noticed in the past as violations of the so-called Hypervirial theorem in Quantum Chemistry). This resolution of paradoxes (essentially the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized continuity equation with a sink term shows up naturally).

DOI
10.48550/arxiv.1605.06534

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We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in an arbitrary parameter-space, in the sense of Berry's adiabatic and cyclic procedures). In this short note, we report modifications of these two theorems when such non-Hermiticities appear, and we demonstrate how they resolve certain Quantum Mechanical paradoxes (most of them having been noticed in the past as violations of the so-called Hypervirial theorem in Quantum Chemistry). This resolution of paradoxes (essentially the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized continuity equation with a sink term shows up naturally).
For real potentials and for arbitrary boundary conditions we point to previously noted paradoxes in applications of the Ehrenfest and Hellmann-Feynman theorems, their resolution and their role on the consistency of quantum mechanical uncertainty relations. The paradoxes originate from a hidden non-Hermitian character of the kinetic energy operator and are resolved if the proper boundary terms (almost always discarded in the literature) are taken seriously. These non-Hermitian contributions (reflections at a deeper level of topological anomalies) follow their own patterns (for any dimensionality, for both Schroedinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities. In continuous nonrelativistic cases, these have the forms that had earlier been used by Chemists to describe atomic fragments of polyatomic molecules - while for Dirac/Weyl or lattice models they appear to have the corresponding relativistic forms. In spite of the fact that these boundary terms originate from a deep mathematical anomaly, we point out examples where such non-Hermiticities have physical significance in Quantum Condensed Matter Physics (for both conventional and topologically nontrivial materials). In all stationary state examples considered, these non-Hermitian boundary terms have turned out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2-D Landau system in an external electric field). Finally, the non-Hermitian terms are also shown to be crucial for the consistency of the standard uncertainty relations (of Kennard/Robertson-type) in multiply-connected space or, generally, in any system that satisfies a Bloch theorem.
For real potentials and for arbitrary boundary conditions we point to previously noted paradoxes in applications of the Ehrenfest and Hellmann-Feynman theorems, their resolution and the role played on the consistency of quantum mechanical uncertainty relations. The paradoxes originate from a hidden non-Hermitian character of the kinetic energy operator and are resolved if the proper boundary terms (almost always discarded in the literature) are taken seriously. These non-Hermitian contributions (reflections at a deeper level of topological anomalies) follow their own patterns (for any dimensionality, for both Schroedinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities. In continuous nonrelativistic cases, these have the forms that had earlier been used by Chemists to describe atomic parts of polyatomic molecules while for Dirac/Weyl or lattice models they appear to have the corresponding relativistic forms. In spite of the fact that these boundary terms originate from a deep mathematical anomaly, we point out examples where such non-Hermiticities have physical significance in Quantum Condensed Matter Physics (for both conventional and topologically nontrivial materials). In all stationary state examples considered, these non-Hermitian boundary terms have turned out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2-D Landau system in an external electric field). Finally, the non-Hermitian terms are also shown to be crucial for the consistency of the standard uncertainty relations (of Kennard/Robertson type) in multiply-connected space or, generally, in any system that satisfies the Bloch theorem.
Rarely noted paradoxes in applications of fundamental quantum relations are pointed out, with their resolution leading to emergent non-Hermitian behaviors due to boundary terms – even for closed systems and with real potentials. The role played by these non-Hermiticities on the consistency of quantum mechanical uncertainty relations is discussed, especially in multiply-connected spaces (more generally, for any system that satisfies the Bloch theorem of Solid State Physics). These subtleties – reflections of topological quantum anomalies – follow their own patterns (for any dimensionality, for both Schrödinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities Jg. In continuous nonrelativistic models, these have the forms that had earlier been used by Chemists to describe atomic fragments of polyatomic molecules, while for Dirac/Weyl or other lattice models they have more interesting relativistic forms only recently worked out in graphene models. In spite of the deep mathematical origin as quantum anomalies examples are provided here, where such non-Hermiticities have a direct physical significance (for both conventional and topological materials). In all stationary state examples considered, these non-Hermitian boundary terms turn out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2D Landau system in an external in-plane electric field), where some particular Jg seems to be related to the well-known dissipationless edge currents. More generally, the non-Hermitian terms play a subtle role on Berry curvatures in solids and seem to be crucial for the consistent application of the so called Modern Theories of Polarization and Orbital Magnetization. It is emphasized that the above systems can be _closed_ (in multiply-connected space, so that the boundaries disappear, but the non-Hermiticity remains), a case in non-Hermitian physics that is not usually discussed in the literature; it is also stressed that a mapping between the above non-Hermiticity (for continuous systems) and the many recent available results in tight-binding solid state models (leading to the so-called exceptional points) is expected to promote enhanced understanding of quantum behavior at the most fundamental level.
We revisit the celebrated Hellmann-Feynman theorem (HFT) in the parity-time (PT)(\mathcal{PT})(PT) invariant non-Hermitian quantum physics framework. We derive a modified version of HFT by changing the definition of inner product and explicitly show that it holds good for both PT\mathcal{PT}PT broken and unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even the PT\mathcal{PT}PT noninvariant Hamiltonian. We consider several examples of discrete and continuum variable systems to test our results. We find that, if the eigenvalue goes through a real-to-complex transition as a function of the Hermiticity breaking parameter, both sides of the modified HFT expression diverge at that point. If that point turns out to be an EP of the PT\mathcal{PT}PT-invariant quantum theory, then one also sees the divergence at EP. Moreover, we have also demonstrated that using the modified HFT can give rise to a potential numerical advantage for computing the expectation value of many-body operators for interacting many-body Hamiltonian. Finally, we also derive a generalized virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.
We revisit the celebrated Hellmann-Feynman theorem (HFT) in the PT invariant non-Hermitian quantum physics framework. We derive a modified version of HFT by changing the definition of inner product and explicitly show that it holds good for both PT broken, unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even PT non-invariant Hamiltonian. We consider several examples of discrete and continuum systems to test our results. We find that if the eigenvalue goes through a real to complex transition as a function of the Hermiticity breaking parameter, both sides of the modified HFT expression diverge at that point. If that point turns out to be an EP of the PT invariant quantum theory, then one also sees the divergence at EP. Finally, we also derive a generalized Virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.
Abstract Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.