  In this paper we study the quantization of the nonlinear oscillator
introduced by Mathews and Lakshmanan. This system with position-dependent mass
allows a natural quantization procedure and is shown to display shape
invariance. Its energy spectrum is found by factorization. The linear harmonic
oscillator appears as the $\lambda\to 0$ limit of this nonlinear oscillator,
whose energy spectrum and eigenfunctions are compared to the linear ones.
