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### Analysis of Harmonic Oscillator [duplicate]

In griffiths, Why was the lowest chosen like that while in the ladder analogy, the lower energies are like $a_-^n\psi$? That is, why didnt he chose the lowest rung to be $$a_-^n\psi$$
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### Origin of Ladder Operator methods

Ladder operators are found in various contexts (such as calculating the spectra of the harmonic oscillator and angular momentum) in almost all introductory Quantum Mechanics textbooks. And every book ...
### What does periodicity of $e^{-iHt/\hbar}$ mean in physical terms?
The unitary time evolution operator $U(t)=e^{-iHt/\hbar}$ has some distinct flavour of periodicity to it because of $e^{x+2\pi i}=e^x$. Is this periodicity reflected in any way in physical systems? ...