The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.

中文翻译：

---

从 QED 推导薛定谔方程

薛定谔方程将波函数和电势的涌现量联系起来，并被假定为量子物理原理或通过启发式获得。然而，物理一致性要求薛定谔方程是一个低能动力学条件，我们可以从量子电动力学的基础中推导出来。由于电磁耦合常数值较小，我们表明电势准确地代表了中间低能光子交换的贡献。然后，从总的非相对论能量关系中，我们看到电子波函数的主导项是满足薛定谔方程的平面波的叠加。我们的推导表明，薛定谔方程不是能量守恒关系，因为它的中间项并不代表假设的电子动能。我们分析了薛定谔方程的物理内容，并通过计算和评估氢原子基态的物理量来验证我们的评估。此外，我们解释了为什么非相对论量子动力学与经典动力学不同。本科生可以跟着推导，因为它涉及基本的物理概念和数学表达式，我们解释了每一步。