Semiconductor quantum wells with BenDaniel - Duke boundary conditions: approximate analytical results

The Schrodinger equation for a particle moving in a square well potential with BenDaniel - Duke boundary conditions is solved. Using algebraic approximations for trigonometric functions, the transcendental equations of the bound states energy are transformed into tractable, algebraic equations. For the ground state and the first excited state, they are cubic equations; we obtain simple formulas for their physically interesting roots. The case of higher excited states is also analyzed. Our results have direct applications in the physics of type I and type II semiconductor heterostructures.