Multidimensional fission model with a complex absorbing potential

We study the dynamics of multidimensional quantum tunneling by introducing a complex absorbing potential into a two-dimensional model for spontaneous fission. We first diagonalize the Hamiltonian with the complex potential to determine a resonance state as well as its lifetime. We then solve the time-dependent Schrödinger equation with such basis in order to investigate the tunneling path. We compare this method with the semiclassical method for multidimensional tunneling with imaginary time. A good agreement is found both for the lifetime and for the tunneling path.

Figure 1

(a) One-dimensional potential $U\left(R\right)$ with $R_b$ = 6 fm (the red solid line). The figure also shows the modified potential for the initial wave function (the blue dashed line) and the absorbing potential $W\left(R\right)$ (the green dotted line). (b) The resultant initial wave function (the blue dashed line) and the resonance wave function (the red solid line).

Figure 2

Time evolution of the square of the wave function during the decay process. The time is indicated in unit of fm/$c$.

Figure 3

Survival probability as a function of time computed dynamically using the formula (14) with $R_{\lim }=20$ fm (the blue crosses). The $R_b$ parameter in the potential is taken to be $R_b=11$ fm. The survival probability is plotted both in the linear scale (the upper panel) and in the logarithmic scale (the lower panel). For a comparison, the figure also shows the survival probabilities for an exponential decay with the decay width obtained from the semiclassical approximation given by Eq. (9) (the red solid line).

Figure 4

Wave function for the metastable state plotted in the logarithmic scale. This state corresponds to an eigenstate of the Hamiltonian $H^{\prime }$ for the potential with $R_b=6$ fm and with the imaginary potential given by Eq. (13) with $W_0=-0.1$ MeV and $R_a=11$ fm. The total potential $V(R,\xi )$ is also plotted by the contour lines.

Figure 5

Time evolution of the two-dimensional wave function at time $t=0$, 250, 500, and 1250 fm/$c$. For a comparison, the semiclassical tunneling path is also shown with the dashed line.

Figure 6

Comparison of the quantum mechanical flux computed with Eqs. (15) and (16) (the red arrows) with the semiclassical escape path (the blue dashed line). The two-dimensional potential energy surface is also shown by the black contour lines.

Figure 7

Fission lifetime as a function of the $R_b$ parameter in the potential $U\left(R\right)$. The lifetime computed quantum mechanically is denoted by the blue crosses, while that evaluated with the semiclassical method using Eq. (17) is denoted by the red line.