This paper applies the matrix finite-difference method to the numerical solution of the Schrödinger equation on a discrete spatial grid. The method enables controllable accuracy in calculating the bound-state energy eigenvalues and corresponding eigenfunctions for a prescribed potential. It can be implemented with only a few lines of code in common high-level programming languages, making it accessible and computationally efficient. More importantly, its logic and underlying principles are transparent, helping undergraduate students understand the essential idea of solving the Schrödinger equation numerically and establish a direct connection between differential equations and linear algebra. As examples, the energy eigenvalues and eigenfunctions of the one-dimensional harmonic oscillator are calculated, and the radial wave functions and energy levels of the hydrogen atom are obtained. The matrix finite-difference method is also compared with the widely used matrix Numerov method, with emphasis on their similarities and differences in derivation, accuracy, and applicability. This work aims to demystify the “black box” of numerical computation in quantum mechanics and provide students with a clear and intuitive understanding of the algorithm.

Tensors are a very important concept in theoretical physics. The inertia tensor is the first concept of tensors that students encounter in undergraduate courses. This paper attempts to innovate the teaching content of the inertia tensor by reorganizing the teaching content of the inertia tensor of rigid bodies. It starts with calculating the angular momentum of a rigid body rotating about a fixed point, then introduces the matrix form, and finally introduces the concept of the inertia tensor, dyadic notation, and their calculation rules. This approach helps students understand that the introduction of tensors is natural and is just a mathematical form of problem description, but it brings convenience. Using this as a starting point, it lays a solid foundation for students' subsequent learning of theories involving tensors and also cultivates their innovative thinking.

The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.

Dimension is an important concept in physics learning and research. In dealing with complicated physical problems, dimensionality reduction methods are often used to simplify the problem, so as to analyze and solve the simplified problem in a low dimensional space. However, when faced with specific physical problems, sometimes increasing the dimensions of the system can make the problem become more intuitive and simpler. This paper takes several specific physical problems as examples such as Gaussian integral, determinant calculation, normalizing the eigenstates of the harmonic oscillator in quantum mechanics, and generating functions of special functions, to demonstrate how to effectively solve the difficulties encountered in low dimensional teaching by improving the dimensions of the system. The purpose of this paper is to emphasize the significance of the concept of “dimension reduced strategy” in physics teaching, and cultivate students' divergent thinking abilities.