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2018 | OriginalPaper | Chapter

Towards the Quantum Geometry of Saturated Quantum Uncertainty Relations: The Case of the (Q, P) Heisenberg Observables

Author : Jan Govaerts

Published in: Mathematical Structures and Applications

Publisher: Springer International Publishing

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Abstract

This contribution to the present Workshop Proceedings outlines a general programme for identifying geometric structures—out of which to possibly recover quantum dynamics as well—associated with the manifold in Hilbert space of the quantum states that saturate the Schrödinger–Robertson uncertainty relation associated with a specific set of quantum observables which characterise a given quantum system and its dynamics. The first step in such an exploration is addressed herein in the case of the observables Q and P of the Heisenberg algebra for a single degree of freedom system. The corresponding saturating states are the well-known general squeezed states, whose properties are reviewed and discussed in detail together with some original results, in preparation of a study deferred to a separated analysis of their quantum geometry and of the corresponding path integral representation over such states.

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previous chapter The Yukawa Model in One Space - One Time Dimensions

next chapter The Role of the Jacobi Last Multiplier in Nonholonomic Systems and Locally Conformal Symplectic Structure

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Robertson extended this statement to an arbitrary number of observables in terms of the determinant of their covariance matrix of bi-correlations.

Because of the sesquilinear properties of the inner product defined over Hilbert space.

Because of the hermitian and positive definite properties of the inner product defined over Hilbert space.

For this reason no attempt is being made towards a complete list of references to the original literature which relates to many different fields of quantum physics.

Hence the states |q〉, say, are determined up to a q-independent overall global phase factor which remains unspecified, relative to which all other phase factors are then identified accordingly.

All Baker–Campbell–Hausdorff (BCH) formulae necessary for this paper are discussed in Appendix 2.

With \(du_0 d\bar {u}_0\equiv d{\mathrm {Re}}\,u_0\,d{\mathrm {Im}}\,u_0\).

Unless of course, one considers a reference state which itself is again the Fock vacuum for some other Fock algebra constructed out of the Heisenberg algebra of the observables Q and P, as is the case with the squeezed quantum states to be identified in Sect. 4.

Note the identity \(\cosh ^2 2r - \cos ^2\theta \sinh ^2 2r=1+\sin ^2\theta \sinh ^2 2r\).

In the simple situation of the harmonic oscillator of mass m and angular frequency ω, and by choosing then \(\ell _0=\sqrt {\hbar /(m\omega )}\), all these general squeezed states evolve coherently into one another with parameters u ₀ and z whose time dependence is given by u ₀(t) = u ₀ e ^iωt and z(t) = ze ^2iωt.

Note the slight abuse of notation which is without consequence, which consists in denoting as a dependence on z a dependence of (b(z, u ₀), b ^†(z, u ₀)) which is in fact separate in r and in e ^iθ while z = re ^iθ.

In the same way that b(u ₀)| Ω₀(u ₀)〉 = 0, a| Ω₀(u ₀)〉 = u ₀| Ω₀(u ₀)〉 and b(u ₀) = a − u ₀, corresponding to the case with z = 0.

Note that because of (90), one also has the identity D(u ₀)S(z) = S(z)D(u ₀(z)) with \(u_0(z)=\cosh r(u_0-\zeta \bar {u}_0)\). The author thanks Victor Massart for a remark on this point.

Since one has the relations \(\frac {i}{2\lambda _0\hbar }=-\frac {1}{\hbar }\frac {\Delta P}{\Delta Q}e^{-i\varphi }= -\frac {1}{2\ell ^2_0}\frac {1-i\sin \theta \sinh 2r}{\cosh 2r + \cos \theta \sinh 2r}\).

Note that \((-i)[\bar {A},\bar {B}]\) and \(\left \{\bar {A},\bar {B}\right \}\) are hermitian (or even self-adjoint if A and B are self-adjoint) operators whose expectations values are thus real.

It suffices to consider the generating operator in λ, e ^λA Be ^−λA, expanded in series in λ.

As a reminder we have \(\int _0^1dt\,(1-t)^n\,t^m=n!\,m!/(n+m+1)!\) as well as \(\int _0^1 dt\,t^n=1/(n+1)\), hence in particular \(\frac {d}{d\lambda }e^{A(\lambda )}=\int _0^1dt\, e^{(1-t)A(\lambda )}\frac {dA(\lambda )}{d\lambda } e^{tA(\lambda )}\).

Note that \(\Psi (e^{-x})=1/\Phi (x)=x/(e^x-1)=\sum _{n=0}^\infty B_n x^n/n!\) is a generating function for Bernoulli numbers, B _n. The author thanks Christian Hagendorf for pointing this out to him.

Note that for all practical purposes the results of [15] remain valid as stated if the term \(c\mathbb {I}\) stands for an operator which commutes with both A and B.

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M.M. Nieto, The discovery of squeezed states – in 1927. Preprint LA-UR-97-2760. e-print arXiv:quant-ph/9708012

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Title: Towards the Quantum Geometry of Saturated Quantum Uncertainty Relations: The Case of the (Q, P) Heisenberg Observables
Author: Jan Govaerts
Publisher: Springer International Publishing
Book: Mathematical Structures and Applications
Print ISBN: 978-3-319-97174-2

Electronic ISBN: 978-3-319-97175-9

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-97175-9_11

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