Abstract:

Moving particle in the one-dimensional infinite square potential well is a quantum ideal model.In the traditional textbooks，the boundary condition of the potential well is particularly taken.Either it is symmetric about the coordinate origin point，or the left boundary value of the well is at the origin of the coordinate.First，it is presented how to solve the energy eigenvalues and eigenfunctions for this model with the arbitrary boundary condition by virtue of the three kinds of approaches.Moreover，the results obtained by the different methods are equivalent to each other.Then the derived results are discussed and analyzed.Furthermore，the general formulas for solving the energy eigenvalues and eigenstates for the one-dimensional infinite square potential well with a arbitrary boundary are acquired.It is easy to see that the two physical quantities are dependent on the well width，and the eigenfunctions are related to the boundary value.Finally，the one-dimensional results are extended to two and threedimensional infinite square potential wells with the arbitrary boundaries.

Key words: infinite square potential well')"> infinite square potential well, stationary state')">stationary state, Schr?dinger equation')">Schr?dinger equation, eigenvalues problem')">eigenvalues problem, standing wave ')">standing wave