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arXiv:1301.0923

Fermi's Trick and Symplectic Capacities: A Geometric Picture of Quantum States

Maurice A. de Gosson, Serge M. de Gosson

Published Jan 5, 2013• 0 citations
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Abstract

We extend the notion of quantum blob studied in previous work to excited states of the generalized harmonic oscillator in n dimensions. This extension is made possible by Fermi's observation in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction Ψ or by a certain function g_{F} on phase space canonically associated with Ψ. We study Fermi's function when Ψ is a Gaussian (generalized coherent state). A striking result is that we can use the Ekeland--Hofer symplectic capacities to characterize the Fermi functions of the excited states of the generalized harmonic oscillator, leading to new insight on the relationship between symplectic topology and quantum mechanics.

DOI
10.48550/arxiv.1301.0923

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We extend the notion of quantum blob studied in previous work to excited states of the generalized harmonic oscillator in n dimensions. This extension is made possible by Fermi's observation in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction Ψ or by a certain function g_{F} on phase space canonically associated with Ψ. We study Fermi's function when Ψ is a Gaussian (generalized coherent state). A striking result is that we can use the Ekeland--Hofer symplectic capacities to characterize the Fermi functions of the excited states of the generalized harmonic oscillator, leading to new insight on the relationship between symplectic topology and quantum mechanics.
Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction ΨΨΨ or by a certain function gFg_FgF​ on phase space canonically associated with ΨΨΨ. In this Note we study we study Fermi's function when ΨΨΨ is a squeezed coherent state. We relate it with the Wigner transform of ΨΨΨ, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid gF(x,p)≤0g_{F}(x,p)\leq0gF​(x,p)≤0 is bounded by h/2h/2h/2 and nh/2nh/2nh/2 (n the number of degrees of freedom).
Fermi observed in 1930 that the state of a quantum system may be defined in two different (but equivalent) ways, namely by its wavefunction Ψ\PsiΨ or by a certain function gFg_FgF​ on phase space canonically associated with Ψ\PsiΨ. In this Note we study we study Fermi's function when Ψ\PsiΨ is a squeezed coherent state. We relate it with the Wigner transform of Ψ\PsiΨ, thus generalising a previous observation of Benenti and Strini. We show that the symplectic capacity of the phase space ellipsoid gF(x,p)≤0g_{F}(x,p)\leq0gF​(x,p)≤0 is bounded by h/2h/2h/2 and nh/2nh/2nh/2 (n the number of degrees of freedom).
Abstract I nspired by our work with Basil Hiley, we continue the exploration of Fermi’s Ansatz. Back in 1930, Fermi showed that a wavefunction could be associated with a hypersurface in phase space. Fermi’s approach therefore leads to a geometric picture of quantum states that is based on Fermi ellipsoids. Our investigations with Basil Hiley revealed that the internal energy provided by Bohm’s quantum potential sustains these Fermi blobs whenever a quantum process is unfolding. Moreover, this internal energy is associated with a noncommutative quantum phase space (coarse-grained by quantum blobs - minimum uncertainty phase-space cells that are in a bijective correspondence with the squeezed coherent states familiar from quantum optics). In this paper, we use Fermi’s method in order to gain further insights into the invariance of quantum blobs under canonical transformations. Starting with a real Gaussian, we multiply it by a complex exponential (“chirp”) and thereby obtain a generalized coherent state. The associated change of gauge enables us to make contact with Fermi’s approach. As in de Broglie-Bohm theory, the phase and magnitude of the wavefunction feature explicitly here and are revealed to play a key role. Subsequently, we find the Fermi Hamiltonian operator and associated eigenvalue equation which the generalized coherent state satisfies. Considering Hamilton’s equations written in compact form, we show that the Hamiltonian flow determined by the Fermi Hamiltonian consists of symplectic matrices that belong to a one-parameter subgroup of Sp( n ) - we call this subgroup the canonical group of the generalized coherent state. The lift of this canonical group to the metaplectic group is found to leave the generalized Gaussian invariant; whereas the canonical group itself leaves the associated phase-space quantum blob invariant. We conclude that Fermi’s Ansatz provides an instructive method for proving these invariance results. It sheds light on subtle aspects of quantum processes that other approaches might miss.
Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries. Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture. This allows us to propose a substitute for phase space in quantum mechanics. We study the relationship between quantum blobs with a certain class of level sets defined by Fermi for the purpose of representing geometrically quantum states.
Quantum blobs are the smallest units of phase space that are compatible with the Robertson-Schrödinger indeterminacy relation and invariant under general symplectic transformations [1]. In this article, the distinguishability between pairs of quantum blobs, as measured by quantum fidelity, is defined on complex phase space. Fidelity is physically interpreted as the probability that the pair are mistaken for each other upon a measurement. The mathematical representation is based on the concept of symplectic capacity in symplectic topology. The fidelity is the absolute square of the complex-valued overlap between the symplectic capacities of the pair of quantum blobs. The symplectic capacity for a given quantum blob, onto any conjugate plane of degrees of freedom, is postulated to be bounded from below by the Gromov width h/2. This generalizes the Liouville theorem in classical mechanics, which states that the volume of a region of phase space is invariant under the Hamiltonian flow of the system, by constraining the shape of the flow. It is shown that for closed Hamiltonian systems, the Schrödinger equation is the mathematical representation for the conservation of fidelity.
In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in configuration space. To each of these states, we associate a classs of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.