Considerations on the probability density in complex space

A probability density is proposed in complex space. This probability density is defined by replacing $x$ with $z$ in the Born probability density of quantum mechanics, and it is real valued and non-negative in complex space. In addition, the Eulerian rate equation for the probability density is derived, and the rate of change of the probability density is determined by the total energy density. Furthermore, this complex-extended Born probability density correctly reflects information of the trajectory dynamics for the wave function in complex space.

Figure 1

(Color online) The probability density ${\rho }_{\mathrm{B}}(z,t)$ (solid line), the time derivative of ${\rho }_{\mathrm{B}}(z,t)$ (dotted-dashed line), and the imaginary parts of the local kinetic energy density $(2∕\hslash )\mathrm{Im}\left[K_{\rho }(z,t)\right]$ (short-dashed line) and the potential energy density $(2∕\hslash )\mathrm{Im}\left[V_{\rho }(z,t)\right]$ (long-dashed line) at $z=0.2+1.5i$ for the nonstationary state given in Eq. (15).

Figure 2

(Color online) The Pólya vector field of the QMF and contours of ${\rho }_{\mathrm{B}}(z,t)$ (solid lines) with the stagnation points (squares) and the node (dot) for the nonstationary state given in Eq. (15) at $t=2$. An additional node from the probability density ${\rho }_{\mathrm{P}}(z,t)$ is shown as a triangle.