The mesoscopic nonlinear electro-quantum origin of the Langmuir law

After a classical derivation of the Langmuir law, we perform a simple electro-quantum analysis, based on the Schrödinger and Gauss–Poisson equations. It is shown that, for any stationary state, the approximating solution to the Gauss–Poisson equation, for the large argument expansion of the Hankel function modulus, turns into the Langmuir-like functional dependence. In practice, the different multiplying coefficient, 3/4 instead of 4/9, makes no essential contribution, for it can be incorporated into the actual perveance of the employed vacuum-based diode. Nevertheless, within an improved electro-quantum model, we deal with the (highly) nonlinear Gauss–Schrödinger system. Consequently, not only one obtains the precise form of the Langmuir law, but also it emphasizes some technicalities in the WKB approximation related to the role of the strong electro-quantum (geometry) condition, which must be fulfilled by the cathode-to-anode (separation) distance, in order to obtain the diode in a principally 1 − 10−n confidence Langmuir-regime.