The electronic states in a cylindrical quantum well (CQW) in the presence of a strong axial magnetic field are investigated numerically. The authors have found that the size quantisations due to the lateral confinements may be destroyed by the strong magnetic field and, when the applied field is larger than a critical magnetic field, the quantised energy levels associated with the radial quantum numbers (n=2, 3, 4, . . .) are completely smeared out, except the first radial quantised energy level associated with n=1. The numerical calculation predicts a critical magnetic field B_c varies as R^-2, where R is the radius of the CQW. Using the results obtained, they have successfully explained an unexpected experimental phenomenon related to the damping of oscillations of the density of states in a quasi-one-dimensional wire under a strong longitudinal magnetic field.