The quantum mechanical flux

In a classical system of moving particles the magnitude of the flux vector is the number of particles going per unit time through a unit area perpendicular to that vector. If p(r) and v(r) are the density and average speed of particles at point r, the flux is given by 

It can be written as a classical dynamical variable, that is, a function of positions and momenta of all particles in the system, in the form 

The quantum analog of this observable should be an operator. To find it we start for simplicity in one dimension and consider the time-dependent Schrodinger equation and its complex conjugate 

We next integrate this equation between two points, xi and X2. 4 (x, 1)4 (x, t)dx is the probability at time t to find the particle in x... x -b dh, hence the integral on the left yields the rate of change of the probability P1-2 to find the particle in the range between xi and X2. We therefore get 

Problem 2.14. Use the results above to prove that at steady state of a onedimensional system the flux has to be independent of both position and time. 

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Figure 25. Electron-transfer rate the electronic coupling strength at T = 500 K for the asymmetric reaction (AG = —3ffl2, oh = 749 cm ). Solid line-present full dimensional results with use of the ZN formulas. Dotted line-full dimensional results obtained from the Bixon-Jortner formula. Filled dotts-effective ID results of the quantum mechanical flux-flux correlation function. Dashed line-effective ID results with use of the ZN formulas. Taken from Ref. [28].

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