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In this paper we consider the initial-boundary value problem for thetime-dependent Maxwell-Schr\"{o}dinger equations, which arises in theinteraction between the matter and the electromagnetic field for thesemiconductor quantum devices. A Crank-Nicolson finite element method forsolving the problem is presented. The optimal energy-norm error estimates forthe numerical algorithm without any time-step restrictions are derived.Numerical tests are then carried out to confirm the theoretical results.

A Crank-Nicolson Finite Element Method and the Optimal Error Estimates for the modified Time-dependent Maxwell-Schr\"{o}dinger Equations