The transformation of a Gaussian beam by a thin lens is an important component of the laser principles course, as it enhances students' understanding of laser optical system design. However, because the course usually omits the general propagation of arbitrary laser beams through linear optical systems, students often raise questions when they try to extend their knowledge. This paper derives, under the second-order moment definition, the waist diameter, waist position, and far-field divergence angle of a general laser beam after passing through a linear optical system represented by a ray-transfer matrix. In addition, we retrace the beam-parameter transformation with the q-parameter method, demonstrating the consistency of the two approaches and thereby reinforcing students' understanding of laser-beam behavior. The general beam-parameter transformation theory can be expected to find direct application in the design of multi-lens beam-transformation systems.

In thermodynamics and statistical physics textbooks, when solving for the wave function of translational particles, one of the infinite-potential-well model and periodic boundary conditions is often arbitrarily chosen for the boundary conditions. Since the number of quantum states per unit energy interval obtained under the two boundary conditions is exactly the same, the specific form of the boundary conditions will not affect the results of statistical physics research. However, the choice of boundary conditions does have some impact on students’ understanding of the specific content of upstream and downstream courses, as some courses involve different conceptual applications. This paper suggests that in the course of thermodynamics and statistical physics, the scope of application of the two different boundary conditions should still be explained in detail.