  The electron states in axially symmetric quantum wires are computed by means
of the effective-mass Schroedinger equation, which is written in cylindrical
coordinates phi, rho, and z. We show that a direct discretization of the
Schroedinger equation by central finite differences leads to a non-symmetric
Hamiltonian matrix. Because diagonalization of such matrices is more complex it
is advantageous to transform it in a symmetric form. This can be done by the
Liouville-like transformation proposed by Rizea et al. (Comp. Phys. Comm. 179
(2008) 466-478), which replaces the wave function psi(rho) with the function
F(rho)=psi(rho)sqrt(rho) and transforms the Hamiltonian accordingly. Even
though a symmetric Hamiltonian matrix is produced by this procedure, the
computed wave functions are found to be inaccurate near the origin, and the
accuracy of the energy levels is not very high. In order to improve on this, we
devised a finite-difference scheme which discretizes the Schroedinger equation
in the first step, and then applies the Liouville-like transformation to the
difference equation. Such a procedure gives a symmetric Hamiltonian matrix,
resulting in an accuracy comparable to the one obtained with the finite element
method. The superior efficiency of the new finite-difference (FDM) scheme is
demonstrated for a few rho-dependent one-dimensional potentials which are
usually employed to model the electron states in free-standing and core-shell
quantum wires. The new scheme is compared with the other FDM schemes for
solving the effective-mass Schroedinger equation, and is found to deliver
energy levels with much smaller numerical error for all the analyzed
potentials. Moreover, the PT symmetry is invoked to explain similarities and
differences between the considered FDM schemes.
