A Study On Quantization Rule And Quantum Transmission By Using The Analytical Transfer Matrix Method

Based on the analytic transfer matrix (ATM) method, we discussed the generalized quantization rule and explored the effect of position-dependent mass on in one-dimensional scattering. Furthermore, we extend the ATM method to solve the issue of the bound energy spectrum of position-dependent mass particle in null potential.The question of solving Schr?dinger equation is non-trivial; to date scientists have proposed various methods to solve the equation. However, only a few classes of potentials are found to be exactly solvable, such as Coulomb potential and Morse potential. Moreover, to the best of our knowledge, there are no analytical methods can obtain the exact eigenenergies and eigenfunctions of arbitrary potentials. The essence of the ATM method is dividing the one-dimensional arbitrary potential in concern into a series of thin layers, whose potential can be considered as a constant potential. The series of step-potentials will be a good approximation of the original potential when the width of each layer tends to 0. The wave function in each layer can be expressed as a linear combination of exponent functions. The transfer matrix can be easily obtained via the connecting condition of the wave function and its derivation on the boundary of each section. Solving the matrix equation, the exact quantization rule for arbitrary potential well can be obtained. It is worthwhile to mentioned that the phase contributed by scattering sub-waves is included in the ATM quantization condition. And the phase shift at the turning point is notπ/ 2 in WKB method but equal toπ.In quantum mechanics, tunneling which refers to the phenomena of particles'ability to penetrate potential barrier, is one of properties of wave-particle duality. People have paid much attention to explore the transmission coefficient of electrons penetrating the barriers, because it's the crucial prerequisite to determinate the current-voltage properties of electronic high-speed devices. With previous results given by the ATM method, the transmission formula for particle passing through the arbitrary potential is obtained and can be applied to study the situation of position-dependent mass, in which the phase contributed by the scattering sub-waves is included. Furthermore, the effect of position-dependent mass on transmission is discussion in detail.The ATM method can't be directly used for null potential, because there are no turning points in null potential. Combining the point canonical transformation (PCT) method, the distribution of position-dependent mass is translated into the effective potential, and bound energy spectrum is obtained. With the calculation for the transmission of a position-dependent mass particle passing through the null potential, the one-to-one correspondence between transmission resonance peak and eigenenergy is found and explained.