Uncertainty Inequality Research Articles

Phase and number

Contrary to the widespread belief that no phase observable exists which is canonically conjugate to the number operator, a selfadjoint operator is given with the right properties to be a firm candidate for the quantum phase. Its spectrum, commutation relations, uncertainty inequalities and classical limit are discussed.

Uncertainty Inequalities for Hankel Transforms

In this paper an uncertainty inequality for Hankel transforms is obtained. Let $\nu > 0$ be fixed. We set \[ d\mu _\nu (x) = c_\nu ^{ - 1} x^{2v} dx,\quad c_\nu = 2^{{{\nu - 1} / 2}} \Gamma (\nu + \frac{1}{2}),\] and \[ {\bf J}_\nu (x) = c_\nu x^{ - \nu + {1 / 2}} J_{\nu - {1 / 2}} (x),\] where $J_{\nu - {1 / 2}} (x)$ is a Bessel function of the first kind of order $\nu - \frac{1}{2}$. We define \[ f^ \wedge (t;\nu ) = \int_0^\infty {f(x)J_\nu (xt)d\mu _\nu (x)} .\] A probability frequency function with respect to $d\mu _\nu $, is defined as a nonnegative function in $L_\nu ^1 (0,\infty )$ with norm one, and the generalized variance of a probability frequency function $F(x)$ is defined by \[ V_\nu [F] = \int_0^\infty {x^2 F(x)d\mu _\nu (x)} .\] Let $f(x)$ belong to $L_\nu ^2 (0,\infty )$ with norm one. By Parseval’s equality $| {f(x)} |^2 $ and $| {f^\wedge (x;v)} |^2 $ can be considered as probability frequency functions. The uncertainty inequality \[ V_\nu \left[ {| {f(x)} |^2 } ]V_\nu [ {| {f^ \wedge (x;\nu )} |^2 } \right] \geqq (\nu + \frac{1}{2})^2 \] is proved, and the constant $(\nu + \frac{1}{2})^2 $ is shown to be the best possible.

Measurements of neutral gas density in the working range of the PR-5 system: Yu N Kotel'nikov and G P Kutukov, Zh Tekh Fiz, 38 (5), May 1968, 818–822 ( in Russian).

An uncertainty inequality for the Fourier–Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker–Shannon–Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939–2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.

Improvement in the sensitivity of mass spectrometer MKh-1303: A A Kravchenko, Zavodsk Lab, 34 (5), 1968, 622–624 ( in Russian).

An uncertainty inequality for the Fourier–Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker–Shannon–Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939–2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.