Hi everyone, I was going through the derivation of the first Hohenberg-Kohn theorem (see here under eqn 1.31 for reference) when I noticed a once-obvious statement that didn't seem so obvious anymore. Namely, the proof requires that if you have two Hamiltonians ##H_1 \neq H_2##, then their ground states are not equal ##\Psi_1 \neq \Psi_2##. Maybe this statement is just completely obvious and I'm missing some PDE or linear algebra uniqueness theorem that I should have remembered from college, but for the life of me, I can't prove it. I know that 2nd order ODE's have unique solutions (given the proper assumptions), but why can't we have:(adsbygoogle = window.adsbygoogle || []).push({});

$$H_1\Psi = E_1\Psi$$

$$H_2\Psi = E_2\Psi$$

where ##H_i## is a partial differential operator?

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# I Help with Hohenberg-Kohn Theorem

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