论文标题
建立对Dirac $δ$功能的信心
Building Confidence in the Dirac $δ$-function
论文作者
论文摘要
在本说明中，我们提供了一个示例，旨在突出dirac $δ$功能的多功能性。也就是说，我们计算以三角波函数$ψ（x）$描述的状态下的自由粒子的期望值。由于$ψ（x）$的第一个导数是分段恒定的，并且由于该哈密顿量与二阶空间衍生产品成正比，因此学生通常最终发现期望值为零 - 一个非物理的答案。此问题提供了Dirac $δ$功能的教学应用。通过通过替代途径达到相同的结果，此练习增强了学生对Dirac $δ$功能的信心，并突出了其效率和优雅。
In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac $δ$-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function $ψ(x)$. Since the first derivative of $ψ(x)$ is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac $δ$-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac $δ$-function and highlights its efficiency and elegance.
