General series solution for finite square-well energy levels for use in wave-packet studies

Abstract

We develop a series solution for the bound-state energy levels of the quantum-mechanical one-dimensional finite square-well potential. We show that this general solution is useful for local approximations of the energy spectrum ͑which target a particular energy range of the potential well for high accuracy͒, for global approximations of the energy spectrum ͑which provide analytic expressions of reasonable accuracy for the entire range of bound states͒, and for numerical methods. This solution also provides an analytic description of dynamical phenomena; with it, we compute the time scales of classical motion, revivals, and super-revivals for wave-packet states excited in the well.

Quantum wave packet revivals

“…For example, the energy eigenvalues, E(n x , n y ), and position-space eigenstates, w (nx,ny) (x, y), are given by 2 and w (nx,ny) (x, y) = u (nx) (x)u (ny) (y) (234) where n x , n y = 1, 2, 3, ... are the appropriate quantum numbers and the u n (x) are given by Eqn. (141). The two non-vanishing revival times are given by Eqn.…”

Square wells, quantum wells and ultra-thin metallic films

“…The coefficients q 7 , ...q 16 , of the exact series expansion (53) q 7 (b) = − 1 + 5 · 7 2 · 3 b 2 + 7 · 37 2 3 · 3 · 5 b 4 + 5 2 7 · 7 b 6 (A9) q 8 (b) = 1 + 2 2 · 7 3 b 2 + 2 · 7 2 3 · 5 b 4 + 2 4 5 · 7 b 6 (A10) q 9 (b) = − 1 + 2 · 7b 2 + 7 · 47 2 2 · 5 b 4 + 3229 2 2 · 3 2 · 5 · 7 b 6 + 5 · 7 2 7 · 3 2 b 8 (A11) q 10 (b) = 1 + 2 2 · 5b 2 + 2 · 7 · 13 5 b 4 + 2 4 · 41 3 2 · 7 b 6 + 2 7 3 2 · 5 · 7 b 8 (A12) q 11 (b) = − 1 + 5 · 11 2 b 2 + 7 · 11 · 19 2 2 · 5 b 4 + 11 · 1571 2 3 · 3 2 · 7 b 6 + 11 · 59 · 181 2 7 · 3 2 · 5 · 7 b 8 + 3 2 · 7 2 8 · 11 b 10 (A13) q 12 (b) = 1 + 2 · 5 · 11 3 b 2 + 2 · 11 · 31 5 b 4 + 2 2 · 11 · 139 3 2 · 7 b 6 + 2 3 · 11 · 479 3 4 · 5 · 7 b 8 + 2 8 3 2 · 7 · 11 b 10 (A14) q 13 (b) = − 1 + 11 · 13 3 b 2 + 11 · 13 · 67 2 5 · 5 b 4 + 11 · 13 · 17 · 127 2 2 · 3 2 · 5 · 7 b 6 + 11 · 13 · 23 · 6679 2 7 · 3 4 · 5 · 7 b 8 + + 13 · 211 · 2609 2 7 · 3 2 · 5 2 · 7 · 11 b 10 + 3 · 7 · 11 2 10 · 13 b 12 (A15) q 14 (b) = 1 + 2 · 7 · 13 3 b 2 + 2 · 7 · 11 · 13 5 b 4 + 2 2 · 11 · 13 · 311 3 2 · 5 · 7 b 6 + 2 4 · 11 · 13 · 37 3 4 · 5 b 8 + 2 6 · 13 · 59 3 2 · 5 2 · 11 b 10 + 2 10 3 · 7 · 11 · 13 b 12 (A16) q 15 (b) = − 1 + 5 · 7 · 13 2 · 3 b 2 + 7 2 · 11 2 · 13 2 3 · 3 · 5 b 4 + 11 · 13 · 8521 2 4 · 3 2 · 7 b 6 + 11 · 13 · 79 · 2917 2 7 · 3 4 · 5 b 8 + 7 · 13 · 1206053 2 8 · 3 4 · 5 · 11 b 10 + 17911 · 135721 2 10 · 3 3 · 5 2 · 7 · 11 · 13 b 12 + 11 · 13 2 11 · 5 b 14 (A17) q 16 (b) = 1 + 2 3 · 5 · 7 3 b 2 + 2 2 · 7 · 13 · 41 3 · 5 b 4 + 2 4 · 11 · 13 · 67 3 2 · 5 · 7 b 6 + 2 · 11 · 13 · 2473 3 4 · 5 b 8 + 2 5 · 13 · 4201 3 4 · 5 · 11 b 10 + 2 6 · 266681 3 3 · 5 2 · 7 · 11 · 13 b 12 + 2 11 3 2 · 5 · 11 · 13 b 14 (A18)…”

“…As already mentioned, a finite well only supports a fixed number of bound states. According to several authors (see for instance [16]), this number is given by:…”

A comparative analysis of the photoluminescence spectra of annealed ultrasmall In-rich InGaN/GaN quantum dots and wells

“…where Ee(n) represents the energy levels of conduction (valance) band of the as-grown structure, which is a finite rectangular QW, computed from the formula [21] outlined below:…”