Abstract: The bound states of one-dimensional finite square well cannot be solved exactly. Numerically，the

energy spectrum and its wave function can be obtained from a transcendental equation. In this paper，we solve the

transcendental equation with the Taylor series expansion，which to the first order gives analytical solutions of the energy

spectrum and its wave function. We find that the number of bound states is determined by a dimensionless parameter

R，which is proportional to the width of the well multiplied by the square of the potential height. Except the highest

level wave function，the approximate analytical results show in well agreement with the numerical results. In the

limit of large R，the approximate analytical results recover the exactly solvable problem of an infinite square well.

Key words: one-dimensional finite symmetric square well, the bound state, transcendental equation, energy spectrum, energy eigenwave function