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AM and FM quantum noise properties of semiconductor lasers have been studied theoretically. Theoretical formulations for the AM noise spectrum, photon number probability density, FM noise spectrum, instantaneous frequency probability density, and power spectrum are presented here as functions of semiconductor laser material, structural, and pumping parameters. Two theoretical approaches are employed: one is based on the quantum mechanical Langevin equation, and the other on the density matrix equation. Starting from the quantum mechanical Langevin equation, three different formulations, that is, the rate equation, Fokker-Planck equation, and van der Pol equation, are derived. The parameters which represent stimulated emission, spontaneous emission, and refractive-index dispersion are obtained by using the Kane function interpolated to Halperin-Lax bandtail and the Stern's improved matrix element. The above four different theoretical formulations are related to each other, and the applicability for each method is discussed.