Quantization of a uniform gravitational field with a damping quadratic force

•A quadratic damped mechanical system is quantized using the WKB formalism.•The Hamiltonian of the system has been constructed and used to obtain the Hamilton Jacobi equation.•The quantum results are found to be in exact agreement with the classical results.

A quadratic damped mechanical system is quantized using the WKB formalism. In particular, a uniform gravitational field with a damping force proportional to the square of its speed has been treated in this paper. We construct the Hamiltonian of the system and use it to obtain the Hamilton-Jacobi equation. The action function S has been found using the separation of variables and the chain rule. This function enables us to formulate the wave function and to quantize the system using the WKB approximation. Finally, we are able to prove that the quantum results are in exact agreement with the classical results.