From the equations above, we see that lowest Bohr radius
varies as 1/[m.sup.2], and the orbital frequency as [m.sup.3].
In this regard, theory suggests that, for this to be the case, the radius of the QD should be less than the bulk exciton Bohr radius
, calculated as [a.sub.B] = [epsilon][[??].sup.2]/[mu][e.sup.2], with [epsilon] and [mu] being, respectively, the dielectric constant and the reduced mass of the electron and hole.
where [a.sub.0] is the Bohr radius
and v the electron pulsation (E = h v).
Section 4 will recall some useful inequalities that we shall need and Section 5 focuses on recent applications of Holder's inequality for mixed sums in Functional Analysis and Quantum Information Theory, culminating with the solution of a classical problem from Complex Analysis: the Bohr radius
When the size of semiconductor nanocrystals is smaller than the Bohr radius
of the excited electron-hole pair, quantum confinement effect occurs and the band gap energy starts to increase with the decrease of particle size.
The results obtained in this paper were calculated taking into account the values of the effective "Bohr radius
" [a.sup.*.sub.0] = 9,87nm and the Effective Rydberg constant [R.sup.*.sub.y] = 5,72 meV.
With the introduction of Rydberg scale, the effective Rydberg energy is defined as [R.sup.*] = [m.sup.*.sub.e][e.sup.4]/2[[??].sup.2][[epsilon].sup.2.sub.0] and the effective Bohr radius
as [R.sup.*.sub.B] = [[??].sup.2][[epsilon].sub.0]/[m.sup.*.sub.e][e.sup.2], and hence, the values of these quantities, for GaAs semiconductor, become [R.sup.*] =5.28 meV [R.sup.*.sub.B] = 103 [Angstrom].
Bulk CdSe is a direct bandgap (1.74 eV) II-VI semiconductor with an exciton Bohr radius
of 6 nm [11,12].
Since the uniform circular motion of an electron is in opposition to Heisenberg's Uncertainty Principle (actually [DELTA]r = 0 and [DELTA]mv = 0), my correction to special relativity allows me to consider that when the electron tends to stop, it oscillates around the origin of the x-axis with an amplitude equal to the Bohr radius
and it moves (on average) with twice the minimum speed.
Using the above given parameters of GaAs (x = 0) one can obtain the effective Bohr radius
and the effective Rydberg energy as [??] 101 [Angstrom] and [??] 5.72 meV, respectively.
The constant [a.sub.0], the Bohr radius
, gives the scale of the interaction and [gamma] is the relativistic correction factor.