applied microwave pulses are used to control and measure the state of the transmon qubit. Currently, the design of the microwave pulses for these purposes is done through simple theoretical and/or numerical models that neglect how the transmon can modify the applied microwave field. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrodinger hybrid method for describing the dynamics of a transmon qubit capacitively coupled to a transmission line system. Numerical results are presented using this Maxwell-Schrodinger method to characterize the control and measurement of the state of a transmon qubit. We show that our method matches standard theoretical predictions in relevant operating regimes, and also show that our method produces physically meaningful results in situations where the theoretical models break down. In the future, our method can be used to explore broader operating regimes to search for more effective control and measurement protocols for transmon qubits.