## 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 $$\psi^{\text{ml}}_{x'}$$.

### MSC:

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

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