In the framework of an advanced quantum theory for mesoscopic electric circuits in accord with the discreteness of electric charges, a mesoscopic quartz piezoelectric crystal equivalent circuit is quantized. To resolve the finite difference Schrödinger equation, an improved parameter perturbative method is proposed when WKB and perturbative method are inapplicable. With this method, the energy spectrum and wavefunctions of the system are obtained and used to calculate current quantum fluctuation as an application. The new method would be helpful to the application of the mesoscopic circuits quantum theory. Besides, the detail characters of energy spectrum and wavefunctions in the system would be helpful to the design of integrate circuit.