论文标题
Schwarzschild Spacetime中狄拉克方程的界限：对好奇学生的直觉探索
Bound states of the Dirac equation in Schwarzschild spacetime: an exploration of intuition for the curious student
论文作者
论文摘要
在这项工作中，我们探讨了在施瓦茨柴尔兹柴尔德引力场中量子结合状态的可能性，从而利用了库仑电位中结合状态的基本衍生物的类比，这是量子力学的本科课程中教授的。为此，我们还需要超越非权威主义的量子力学，并利用相对论的狄拉克方程，以实现高级本科或第一年毕业生（特殊）相对论量子力学课程所教的中心潜力。最后，特殊的相对论狄拉克方程必须扩展到弯曲时空的一般相对论版本。所有这些不同的组件作品都以出色的，非常可读性的教科书为书写，并提供了足够的细节，可以让一个好奇的学生学习和探索。 We pull all these threads together in order to explore a very natural question that a student might ask: "If the effective $1/r$ radial potential of the Schwarzschild metric (with angular momentum barrier), as taught in elementary GR courses for undergraduates, appears Newtonian-like (with a $1/r^3$ correction), then is it possible to derive quantum bound states in the Schwarzschild spacetime by simply changing来自$ v_c（r）= - e^2/r $ to $ v_ {schw} = - g m m/r $的径向电势$ v（r）$？
In this work we explore the possibility of quantum bound states in a Schwarzschild gravitational field leveraging the analogy of the elementary derivation of bound states in the Coulomb potential as taught in an undergraduate course in Quantum Mechanics. For this we will also need to go beyond non-relativistic quantum mechanics and utilize the relativistic Dirac equation for a central potential as taught in an advanced undergraduate or first year graduate (special) relativistic quantum mechanics course. Finally, the special relativistic Dirac equation must be extended to the general relativistic version for curved spacetime. All these disparate component pieces exist in excellent, very readable textbooks written for the student reader, with sufficient detail for a curious student to learn and explore. We pull all these threads together in order to explore a very natural question that a student might ask: "If the effective $1/r$ radial potential of the Schwarzschild metric (with angular momentum barrier), as taught in elementary GR courses for undergraduates, appears Newtonian-like (with a $1/r^3$ correction), then is it possible to derive quantum bound states in the Schwarzschild spacetime by simply changing the radial potential $V(r)$ from $V_C(r)=-e^2/r$ to $V_{Schw}=-G M m/r$?"
