Simple derivation of quantum scaling in Child-Langmuir law

A simple derivation of the new scaling of Child-Langmuir law in the quantum regime is presented. Based on a dimensional argument of the Schrodinger equation and the Poisson equation, the limiting current in the deeply quantum regime is found to be proportional to the square root of the gap voltage and to the inverse fourth power of gap spacing. The importance of electron exchange-correlation interactions in the quantum regime is discussed.