Limited To Deep Square Quantum Wire Wave Function

This paper has studied the wavefunction expanded in terms of the two-dimensional harmonic oscillator eigenfunction through calculating the energy of the ground state , the energy of the first excited state and the oscillator strength in a square wire with finite barriers and studied its application in these fields. The most remarkable advantage of this wavefunction is that it can satisfy the continuity of the function and of its derivative divided by the band-mass and it is convenient to calculate some physical magnitudes because the number of the terms is small.It is very important to select a kind of rational wavefuncion for calculating all the physical magnitudes in quantum well wires. For a square quantum wire with finite barriers, there are three kinds of wavefunctions used formerly as follows:1. In 1985, Takeshi Kodama et al. [12] expressed the wavefunction as the combination of the function of the single electron in a one-dimensional square well with the finite barrier to calculate the binding energies of the exciton. This form doesn't satisfy the continuity of the function and of its derivative divided by the band-mass.2. In 1997 and in 1999, A.Sali and M.Fliyou et al. [13,14] selected a kind of wavefunction which can satisfy thecontinuity of the function, but can not satisfy the continuity of its derivative divided by the band-mass.3.In 1996, S.Gangopadhyay et al. [5] expressed the wavefunction in terms of a two-dimensional Fourier series. This form can satisfy the continuity of the function and of its derivative divided by the band-mass. But the number of the terms is so large that it is difficult to calculate the physical magnitudes in the single quantum wire.The first two kinds of wavefunctions are simple formally, but there must be error of the numerical values of some physical magnitudes because there is a trouble with the continuity of the function and of its derivative divided by the band-mass at the boundaries. Though the third kind of wavefunction can satisfy the continuity of the function and of its derivative divided by the band-mass, the number of the terms is so large that it is difficult to calculate the physical magnitudes in the single quantum wire.In this paper, the wavefunction is expanded in terms of the two-dimensional harmonic oscillator eigenfunction and the mismatch of the effective mass is considered. We calculate the energy of the ground state, the energy of the first excited state and the oscillator strength of the single electron in a square quantum wire with finite barriers.From discussing and analyzing the calculated outcomewe obtain the conclusions as follows:1.For the energies of the single electron in a square quantum wire with finite barriers, the former wavefunctions [12-14] are only available to the wide wires. For the oscillator strength of the single electron in a square quantum wire with finite barriers, the former wavefunctions are not appropriate.2. In this paper, the continuity of the wavefunction and of its derivative divided by the band-mass can be satisfied and the number of the terms is small when calculating the energies of the single electron in a square quantum wire with finite barriers, then this wavefunction can also be selected as the envelope function in studying the impurity states and the excitons in the square quantum wires with finite barriers.