Canonical commutation relations on the interval

Lassner, G.; Trapani, Camillo

doi:10.1063/1.527785

Cited by 15 publications

(9 citation statements)

References 3 publications

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“…The same reasoning can be made with any other momentum observable P α , α = 1, only the r.h.s. of (3.22) becomes slightly more complicated [23]. This somewhat long digression should convince the reader that the infinite well problem is really singular, and therefore formal considerations, in particular with respect to boundary conditions, may be misleading (see, for instance, [25] or [26])!…”

Section: )mentioning

confidence: 99%

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Temporally stable coherent states for infinite well and Pöschl–Teller potentials

Antoine

Gazeau

Monceau

et al. 2001

Journal of Mathematical Physics

226

256

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This paper is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Pöschl-Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU (1, 1) symmetryà la Barut-Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.

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Section: )mentioning

confidence: 99%

“…An alternative [23] consists in keeping (3.20) unchanged, but generalizing the usual algebraic formalism to the quasi-*algebra generated by the operators Q, P . By this we mean the following.…”

Section: )mentioning

confidence: 99%

Temporally stable coherent states for infinite well and Pöschl–Teller potentials

Antoine

Gazeau

Monceau

et al. 2001

Journal of Mathematical Physics

226

256

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“…To make the σ-momentum a real quantum mechanical momentum one should check the validity of Canonical Commutation Relations (CCR) of a momentum and a position operator in our system. In general, a problem of CCR on the finite interval is far from being obvious [27]. As opposed to the entire real line case where both positionX and momentumP operators are unbounded, here the operatorX is bounded in the Hilbert space L 2 (0, a) whereas the operatorP = −i ∂ is unbounded.…”

Section: Momentum Seen From the Moving Reference Framementioning

confidence: 99%

Finite size universe or perfect squash problem

Turko

2004

Journal of Mathematical Physics

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We give a physical notion to all self-adjoint extensions of the operator id/dx in the finite interval. It appears that these extensions realize different non-unitary equivalent representations of CCR and are related to the momentum operator viewed from different inertial systems. This leads to the generalization of Galilei equivalence principle and gives a new insight into quantum correspondence rule. It is possible to get transformation laws of wave function under Galilei transformation for any scalar potential. This generalizes mass superselection rule. There is also given a new and general interpretation of a momentum representation of wave function. It appears that consistent treatment of this problem leads to the time-dependent interactions and to the abrupt switching-off of the interaction.

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“…But it is also true for simple quantum mechanical systems. For instance, the ladder operators used in the analysis of the quantum harmonic oscillator, and their related number operator, are all unbounded, [35,39,40]. To deal with these cases, in the past twenty years or so several examples of unbounded operator algebras have been introduced and studied in details.…”

Section: Ii1 O * -Algebrasmentioning

confidence: 99%

Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra

Bagarello

2021

Preprint

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In a recent paper a pair of operators a and b satisfying the equations a † a = bb † + γ1 1 and aa † = b † b + δ1 1, has been considered, and their nature of ladder operators has been deduced and analysed. Here, motivated by the spreading interest in non self-adjoint operators in Quantum Mechanics, we extend this situation to a set of four operators, c, d, r and s, satisfying dc = rs + γ1 1 and cd = sr + δ1 1, and we show that they are also ladder operators. We show their connection with biorthogonal families of vectors and with the so-called D-pseudo bosons. Some examples are discussed.

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Canonical commutation relations on the interval

Abstract: On the Hilbert space ℋ=ℒ 2 (0,L), where (0,L) is a bounded interval of R1, the domain for the canonical commutation relation (CCR) and the CCR quasi-*-algebra is constructed. It is shown that the Bogolubov inequality for a Bose gas (in a box) is fulfilled.

Cited by 15 publications

References 3 publications

Temporally stable coherent states for infinite well and Pöschl–Teller potentials

Temporally stable coherent states for infinite well and Pöschl–Teller potentials

Finite size universe or perfect squash problem

Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra

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