Semi-Classical Approximation in Quantum Mechanics

Маслов, В. П.; Fedoriuk, M. V.

doi:10.1007/978-94-009-8410-3

Cited by 847 publications

(756 citation statements)

References 0 publications

Supporting

Mentioning

750

Contrasting

Unclassified

Order By: Relevance

“…According the general formulation of the classical limit in the framework of orthodox quantum mechanics [Maslov 1981], the classical velocity associated to the particle is given by = ∇ 0 , where 0 is obtained by in the limit 'ℏ → 0 ′ . 4 In the case of a free…”

Section: Appendix: the Classical Limit Of Quantum Mechanics For A Frementioning

confidence: 99%

A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics

Allori

2017

International Studies in the Philosophy of Science

View full text Add to dashboard Cite

In this paper I investigate, within the framework of realistic interpretations of the wave function in nonrelativistic quantum mechanics, the mathematical and physical nature of the wave function. I argue against the view that mathematically the wave function is a two-component scalar field on configuration space. First, I review how this view makes quantum mechanics non-Galilei invariant and yields the wrong classical limit. Moreover, I argue that interpreting the wave function as a ray, in agreement many physicists, Galilei invariance is preserved. In addition, I discuss how the wave function behaves more similarly to a gauge potential than to a field. Finally I show how this favors a nomological rather than an ontological view of the wave function.

show abstract

Section: Appendix: the Classical Limit Of Quantum Mechanics For A Frementioning

confidence: 99%

A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics

Allori

2017

International Studies in the Philosophy of Science

View full text Add to dashboard Cite

show abstract

“…Hence, our problem is reduced to apply Maslov theory (see [9]) to the preceding system. As in [15], we can suppose that supp c q,k,l,γ ⊂ Σ q,l,− with…”

Section: Approximation Of the Evolutionmentioning

confidence: 99%

“…We apply Maslov strategy (see [9]) to the preceding system. Using again Lemma 6.1, it is clear that (y, z) = (y, Z(t, η, y,z, λ q ω), η) is non-focal in the Maslov sense for T 1 < t < T 0 and (y,z, η) ∈ Σ q,l,− so that we can build an approximate solution τ ap under the form…”

Section: Approximation Of the Evolutionmentioning

confidence: 99%

“…Moreover, denoting l 1 (y, z, η, y * , z * , η * ) the sub-principal symbol of L N (h), it follows from Theorem 10.5 in [9] that τ q,k,l,0 (t, y, z, η) = exp Using the fact that the sub-principal symbol p q,1 of P q (h) is null, an easy calculation shows that l 1 vanishes along the Hamiltonian flow so that M (s) is identically zero, and then τ q,k,l,0 (t, y, z, η) = c l (y,z, η).…”

Section: Approximation Of the Evolutionmentioning

confidence: 99%

“…Thanks to Assumption 3 and using Definition 7.4 in [9] of the path index, it is clear that µ q,l is locally constant with respect to (y, η). Moreover, τ q,k,l,j being compactly supported with respect to (y, η), it follows from Theorem 3.15 in [9] that (ih∂ t − L N (h))τ ap = h N +1 r N +1 (t, y, z, η) with r N +1 in S 1/2 (R t × R n y,z,η , h 2 ) and the proof is complete.…”

Section: Approximation Of the Evolutionmentioning

confidence: 99%

See 2 more Smart Citations

Scattering amplitude for the Schrödinger equation with strong magnetic field

Michel¹

2008

Journées Équations Aux Dérivées Partielles

View full text Add to dashboard Cite

show abstract

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

View full text Add to dashboard Cite

A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.When the stationary Born-Oppenheimer description for nearly degenerate state is regarded as "embedded" in a broader dynamic formulation, namely through solution of the time dependent Schrödinger equation, the wave function becomes necessarily complex. The analytic behavior of this function in the complex time-plane can be exploited to gain information about the connection between phases and moduli of the componentamplitudes. One form of this connection are a pair of reciprocal (Kramers-Kronig type) relations in the complex time-domain. These show that certain phase changes in a component amplitude (such as, e.g., lead to a non-zero Berry-phase) require changes in the amplitude-moduli, too, and imply degeneracies of the electronic state.The subject of conical degeneracy, or intersection, of adjacent potential surface is extended to nonlinear nuclear-electronic coupling, which can then result in multiple conical degeneracies on a nuclear coordinate plane. We treat cases of double and fourfold intersections, the latter under trigonal symmetry, and employ an analytic-graphical phase tracing method to obtain the resulting Berry phases as the system circles around some or all of the degeneracy points. These phases can take up values that are all integral multiples of π, with integers that vary with the physical situation (or with the model postulated for it).A further type of invariance, with respect to gauge transformation, is also tied to the complex form of the wave function. For a many-component state this leads to a consideration of tensorial (Yang-Mills type) fields F for molecular systems. It is shown that it is the truncation of the Born-Oppenheimer electron-nuclear superposition that generates the molecular Yang-Mills fields.The equations of motion for the nuclear degrees of freedom are derived from a Lagr...

show abstract

Semi-Classical Approximation in Quantum Mechanics

Cited by 847 publications

References 0 publications

A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics

A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics

Scattering amplitude for the Schrödinger equation with strong magnetic field

Complex States of Simple Molecular Systems

Contact Info

Product

Resources

About