Article published in Mathematics Exchange, 7(1), 2010.In quantum mechanics and in other branches of physics, it is common to approach physical  problems using algebraic and analytic methods. Examples include the use of differential equations for many interesting models, the use of quantum groups in quantum physics, and of differential geometry in relativity theory. In this article, we discuss the Hermite polynomials, some of their properties and a brief description of their applications to the Quantum Harmonic-Oscillator