THE DENSITY MATRIX FOR AN ELECTRON CONFINED IN QUANTUM DOTS UNDER A UNIFORM MAGNETIC FIELD

By using unitary transformation and representation theory of quantum mechanics, we obtain two forms of density matrix respectively in (x₁, p₂) and (x₁, x₂) representations for an electron, with an anisotropic effective mass, confined in quantum dots under a uniform magnetic field (UMF). We find that both forms of density matrix can play an implemental role in the thermodynamical calculation. But, in order to calculate thermal average of angular momentum in z direction for this system, the density matrix in (x₁, p₂) representations is a more convenient form. When the confinement to electron in quantum dots disappears, we encounter a divergence problem of thermal average for some physical quantities. However, in the calculation of the thermal average of Hamiltonian, these divergent quantities cancelled each other out. And we eventually obtain a reasonable result.