Atomic Model of Semiconductor Quantum Dots

As described above, a semiconductor quantum dot can be modelled with good accuracy as a hydrogen-like atom. An excellent introduction to the quantum hydrogen atom is presented in chapter II of volume 43 of Modern Aspects of Electrochemistry, as well as in classic texts. 

Briefly, the electron in a hydrogen atom lies in a spherically symmetric Coulomb potential due to the positively charged nucleus (in atomic units with e = me = h = )  

The Schrodinger equation that describes the stationary states of an electron in this potential is 

Recall the well-known result that symmetries in a problem are associated with conserved quantities. Because of the spherical and reflection symmetries in the potential, it immediately follows that angular momentum and parity are conserved quantities in this system. In spherical coordinates, the above equation is separable into its radial and angular components and the radial equation corresponding to an angular momentum f is 

The eigenvalues En = and eigenvectors fl (r) which are the products of the radial functions - jrndr) and spherical harmonics Yim have long been known. An excellent pictorial representation of the radial waveflinctions can be found in the text Theoretical Atomic 

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