Exact analytic analysis of finite parabolic quantum wells with and without a static electric field

Exact analytic solutions of finite parabolic quantum wells are derived for both the unperturbed and the electric-field-applied cases. Several normalized parameters are defined so as to make our results universal within the scope of the envelope function approximation with a constant effective mass assumed. The Stark resonance position and the width in the electric-field-applied case can be obtained simultaneously from the complex eigenvalue ${\mathit{E}}_0$-iΓ/2 of the system. By comparing the results calculated, respectively, by employing the exact solutions and the infinite-parabolic-well approximation, the validity of the approximation is rigorously examined. It is shown that the infinite-parabolic-well approximation is valid only under certain conditions as discussed in the text.