摘要
Using unitary transformations, this paper obtains the eigenvalues and the common eigenvector of Hamiltonian and a new-defined generalized angular momentum （Lz） for an electron confined in quantum dots under a uniform magnetic field （UMF） and a static electric field （SEF）. It finds that the eigenvalue of Lz just stands for the expectation value of a usual angular momentum lz in the eigen-state. It first obtains the matrix density for this system via directly calculating a transfer matrix element of operator exp（-βH） in some representations with the technique of integral within an ordered products （IWOP） of operators, rather than via solving a Bloch equation. Because the quadratic homogeneity of potential energy is broken due to the existence of SEF, the virial theorem in statistical physics is not satisfactory for this system, which is confirmed through the calculation of thermal averages of physical quantities.
Using unitary transformations, this paper obtains the eigenvalues and the common eigenvector of Hamiltonian and a new-defined generalized angular momentum （Lz） for an electron confined in quantum dots under a uniform magnetic field （UMF） and a static electric field （SEF）. It finds that the eigenvalue of Lz just stands for the expectation value of a usual angular momentum lz in the eigen-state. It first obtains the matrix density for this system via directly calculating a transfer matrix element of operator exp（-βH） in some representations with the technique of integral within an ordered products （IWOP） of operators, rather than via solving a Bloch equation. Because the quadratic homogeneity of potential energy is broken due to the existence of SEF, the virial theorem in statistical physics is not satisfactory for this system, which is confirmed through the calculation of thermal averages of physical quantities.
