Symmetry, Conservation Laws, and Accidental Degeneracy
in Quantum Semiconductor Heterostructures
Symmetry, its identification, and its consequences for the properties of physical systems has been studied for over 150 years. A full appreciation of the level degeneracy in the electronic states in the hydrogen atom showed that the symmetry of the Coulomb potential (1/r) is not just the apparent rotational symmetry (O(3)) with its level multiplicity governed by orbital angular momentum. Rather, the degeneracy associated with different orbital angular momenta is now understood as arising from rotational symmetry in 4D defined by the symmetry group O(4). This insight has led to a deeper understanding of the periodic table of elements.
Such “accidental degeneracies” occur in the 3D quantum harmonic oscillator, where the level degeneracy is governed by SU(3) symmetry. In my work, I have discovered two additional examples of accidental degeneracy and its removal under remarkable conditions based on dynamical considerations – the quantum wire and the cubic quantum dot. This also applies to the degeneracies of the electrodynamic cavity.
A quantitative analysis is presented for the energy levels and wavefunctions of carriers in quantum wires and cubic quantum dots of GaAs embedded in Ga(1−x)AlxAs with finite confining potential barriers at their interfaces. Such systems have non-separable potentials; we use Hermite interpolation schemes in finite element analysis to obtain unprecedented accuracies of 1 part in 10-10-10-14. Some computational tricks that are used will be discussed.
The energy spectrum has substantially reduced degeneracies as compared to the analytically determined energy levels of the infinite-barrier quantum box of the same dimensions. The level degeneracy of states is explained by group representations of the point group Oh. Projection operators for the irreducible representations provide a unique way of obtaining the linear combinations of the degenerate wavefunctions, which form a basis set for the representations. Energy level splittings in the presence of externally applied electric and magnetic fields are also presented. These calculations are relevant to the design of quantum dot lasers, which can be used in LED displays.
Undergraduate and graduate students are especially invited to attend – details of the theory will be made manageable.
Refreshments will be served in Olin Hall 118 at 3:30 P.M.