Confinement limit of the Dirac particle in one dimension

For a particle of mass $m$ that obeys the time-independent Dirac equation in one dimension with a symmetric potential, Unanyan et al. [Phys. Rev. A 79, 044101 (2009)] recently pointed out that the inequality $\Delta x=\sqrt{\langle x^2}\rangle >\lambda /2$ can be derived simply from the Dirac equation. Here $\lambda =\hslash /(\mathit{mc})$ is the Compton wavelength, $x$ is the particle coordinate, and $\langle x^2\rangle$ is the expectation value of $x^2$. We conjecture that a new, more stringent limit $\Delta x⩾\lambda /\sqrt{2}$ holds for any symmetric potential. We present a model analysis on which the conjecture is based.

Figure 1

$\Delta x$ of model I plotted as a function of $a$. The solid and dashed lines are, respectively, for the cases with potentials of Eqs. (13) and (21). The dotted line indicates $\Delta x=\lambda /\sqrt{2}$.

Figure 2

$\Delta x$ of model II plotted for a fixed value of $a=0.1$ and $b=0.005$ (solid line), 0.01 (spaced-dashed line), 0.05 (long dashed line), 0.1 (dashed line), and $0.15$ (dashed-dotted line). Lines for $b<0.005$ are virtually indistinguishable from that of $b=0.005$. $\Delta x$ becomes minimum around $b=0.002$ and $h=0.25$. The dotted line indicates $\Delta x=\lambda /\sqrt{2}$.

Figure 3

Comparison of the densities of models I and II. The solid and dashed lines show, respectively, $\rho (x)$ for models I and II. In both models, $a=0.1$ is assumed. In model II, we assume $b=0.002$ and $h=0.25$. For these values of $b$ and $h$, $\Delta x$ becomes approximately minimum. In addition, $\rho (x)$ of Eq. (11) with $E=0$ is shown with a dotted line.