Square wells, quantum wells and ultra-thin metallic films

Abstract

The eigenvalue equations for the energy of bound states of a particle in a square well are solved, and the exact solutions are obtained, as power series. Accurate analytical approximate solutions are also given. The application of these results in the physics of quantum wells are discussed,especially for ultra-thin metallic films, but also in the case of resonant cavities, heterojunction lasers, revivals and super-revivals.

Quantum Wells and Ultrathin Metallic Films

“…We shall write in a more explicit form Eq. 15, taking into account both the sign of the tan function (or of the cot function, which is, evidently, the same thing), as already mentioned, and the intervals of monotony of the functions sin x=x, cos x=x [13]. The extremum points of the function cos x=x are given by the roots r cn of the equation:…”

Siewert solutions of transcendental equations, generalized Lambert functions and physical applications

“…Somewhat later, Siewert obtains a simpler solution [12]. A very precise approximate analytic solution of (65) was obtained through algebraization [39]; it is useful for the calculation of energy levels in heterojunctions and quantum dots. A more complicated variant of (65) is the Kepler equation for elliptic orbits [1] (e is the excentricity and, in this section only, has nothing to do with the basis of Nepperian logarithms):…”

Approximate expressions for solutions to two kinds of transcendental equations with applications

“…We will try to find approximate solutions as follows. Note that, p p -< < y 2 2 , so higher powers of y could eventually be neglected in equation (11). Keeping the constant, linear and quadratic terms, and neglecting the cubic and quartic ones in equation (11) Taking y 20 as the initial guess value, applying the Newton method to equation (11), iterating one step and noticing that y 20 satisfies equation (14), give the second approximations to the positive roots of equation (11) as a p a p a p = - Finally, based on equation (4), the second approximate expression of the + ( ) n 1 th root of equation (1) is…”

“…However, their results showed large errors. Based on the algebraic approximations of trigonometric functions, it is possible to transform a class of transcendental equations in approximate, tractable algebraic equations [4,11,12]. As the algebraization used in those papers is, to a certain extent, an ad hoc procedure, this approximation must be used with a certain caution in order to avoid the appearance of spurious roots or of roots with too large errors [12].…”