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

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 \(\psi^{\text{ml}}_{x'}\).


46N50 Applications of functional analysis in quantum physics
81P15 Quantum measurement theory, state operations, state preparations
Full Text: DOI arXiv


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