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For simplicity, suppose we are only talking about discrete energy levels, ie, bound state case. The energy levels are $E_1, E_2\cdots$, and the corresponding wave functions are $\psi_1, \psi_2 \cdots$.
My question is, is it true that $\sigma_x \sigma_p$ is minimum when $n=1$ for the eigenstates.
I came across question because I found harmonic oscillator and infinite potential well problems satisfy this statement, so I want to know if this is a general case.
I think this may be true because for ground state, there is no node ground state wave function. Thus the $\sigma_p$ may be small compare to other eigenstates.