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The solutions of the Schroedinger equation are complex-valued, so your inner product needs to be $$(u,v) = \int_\Omega \bar u(x) v(x)\; dx,$$ and the norms then become $$\|u\| = (u,u)^{1/2}.$$ But it is also worth pointing out that you shouldn't think of the Schroedinger equation as a variation of the heat equation that just happens to have an ...

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