perturbation emerges from the interaction of the two subsystems rather than an external potential of the standard Rayleigh–Schrödinger perturbation theory. Each of the wave functions for the electrons of the sample (S) and the tip (T) decay into the vacuum after hitting the surface potential barrier, roughly of the size of the surface work function. The wave functions are the solutions of two separate Schrödinger's equations for electrons in potentials US and UT. When the time dependence of the states of known energies and is factored out, the wave functions have the following general form If the two systems are put closer together, but are still separated by a thin vacuum region, the potential acting on an electron in the combined system is UT + US. Here, each of the potentials is spatially limited to its own side of the barrier. Only because the tail of a wave function of one electrode is in the range of the potential of the other, there is a finite probability for any state to evolve over time into the states of the other electrode. The future of the sample's state μ can be written as a linear combination with time-dependent coefficients of and all , with the initial condition . When the new wave function is inserted into the Schrödinger’s equation for the potential UT + US, the obtained equation is projected onto each separate (that is, the equation is multiplied by a and integrated over the whole volume) to single out the coefficients . All are taken to be nearly orthogonal to all (their overlap is a small fraction of the total wave functions), and only first order quantities retained. Consequently, the time evolution of the coefficients is given by . Because the potential UT is zero at the distance of a few atomic diameters away from the surface of the electrode, the integration over z can be done from a point zo somewhere inside the barrier and into the volume of the tip (z>zo). If the tunneling matrix element is defined as , the probability of the sample's state μ evolving in time t into the state of the tip ν is . In a system with many electrons impinging on the barrier, this probability will give the proportion of those that successfully tunnel. If at a time t this fraction was , at a later time t+dt the total fraction