Maximal localization in the presence of minimal uncertainties in positions and in momenta. (English) Zbl 0883.46049

Hinrichsen, Haye; Kempf, Achim

J. Math. Phys. 37, No. 5, 2121-2137 (1996).

The authors continue their study of representations of the ‘corrected’ canonical commutation relations \[ [x,p]= i\hbar(1+\alpha x^2+\beta p^2), \] where \(\alpha\geq 0\) and \(\beta\geq 0\). In particular, they study a representation on a generalized Fock space and calculate a maximum position localization state \(\psi^{\text{ml}}_x\) that has the properties \[ \langle\psi^{\text{ml}}_x, x\psi^{\text{ml}}_x\rangle= x,\;\langle\psi^{\text{ml}}_x, p\psi^{\text{ml}}_x\rangle= 0,\;(\Delta x)_{\psi^{\text{ml}}_x}=\Delta x_{\min}, \] the minimum value of the uncertainty in \(x\). The authors use a generalization of Rodriguez’s formula to calculate the properties of the coefficients in the expansion of \(\psi^{\text{ml}}_x\) in terms of the Fock basis, and the transition probabilities between two different states \(\psi^{\text{ml}}_x\) and \(\psi^{\text{ml}}_{x'}\).

Reviewer: C.M.Edwards (Oxford)

MSC:

46N50	Applications of functional analysis in quantum physics
81P15	Quantum measurement theory, state operations, state preparations

Keywords:

corrected canonical commutation relations; generalized Fock space; maximum position localization state; Rodriguez’s formula; Fock basis; transition probabilities

References:

[1]	DOI: 10.1103/PhysRevD.15.2795
[2]	DOI: 10.1016/0375-9601(94)90838-9
[3]	DOI: 10.1016/0370-2693(89)91366-X
[4]	DOI: 10.1016/0370-2693(90)91927-4
[5]	DOI: 10.1103/PhysRevD.49.5182
[6]	DOI: 10.1103/PhysRevD.49.5182
[7]	DOI: 10.1142/S0217751X95000085
[8]	DOI: 10.1007/BF00420513 · Zbl 0771.17012
[9]	DOI: 10.1063/1.530204 · Zbl 0796.17016
[10]	Kempf A., Proc. XXII DGM Conf., Sept. 93, Adv. Appl. Cliff. Alg. (Proc. Suppl.) 1 pp 87– (1994)
[11]	DOI: 10.1063/1.530798 · Zbl 0877.17017
[12]	DOI: 10.1007/BF01690456
[13]	DOI: 10.1103/PhysRevD.52.1108
[14]	DOI: 10.1016/0370-2693(92)91044-A
[15]	DOI: 10.1007/BF00762790 · Zbl 0806.17018
[16]	Koornwinder T. H., Trans. AMS 333 pp 445– (1992)
[17]	DOI: 10.1016/0370-2693(94)90940-7
[18]	DOI: 10.1063/1.530644 · Zbl 0826.17018
[19]	Faddeev L. D., Alg. Anal. 1 pp 178– (1989)