We present a full algebraic derivation of the wavefunctions of the simple harmonic oscillator in coordinate and momentum space. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the representation-dependent wavefunctions from the representation-independent energy eigenfunctions. It is simple to incorporate into the undergraduate and graduate curricula. This new derivation begins with the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schroedinger factorization method), and then supplements it by employing the translation (or boost) operator to determine the wavefunctions algebraically, without any derivatives. In addition, we provide a summary of the history of this approach, which seems to have been neglected by most historians of quantum mechanics, until now.

中文翻译：

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简谐振子的完全代数解

我们提出了坐标和动量空间中简谐振子的波函数的完整代数推导。该推导通过从与表征无关的能量本征函数完成与表征相关的波函数的推导，说明了简谐振子的抽象方法。很容易融入本科和研究生课程。这种新的推导开始于 Dirac 在 1947 年首次提出的标准方法（这里根据 Schroedinger 分解方法的精神稍作修改），然后通过使用平移（或提升）算子以代数方式确定波函数来补充它，没有任何衍生。此外，我们还总结了这种方法的历史，