Real Solutions in Time Independent Schrodinger's Equation?

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The discussion centers on the time-independent Schrödinger equation (TISE) and the implications of a real Hamiltonian operator (H-hat) on the nature of its solutions. It is established that if H-hat is a compact self-adjoint operator on a complex separable Hilbert space, the solutions u(x) can be real functions. This conclusion is supported by the properties of the spectral equation, which is a PDE/ODE with real coefficients, allowing for a basis of real functions in the solution space. However, it is acknowledged that solutions can still be complex due to the nature of the Hilbert space.

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spaghetti3451
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This question is about the time independent Schrödinger's equation, but is best posted here.

In the TISE, all the terms in H-hat are real, so it is possible, and not uncommon, for the solutions u(x) to also be purely real.

I don't understand why H-hat is real implies that u (x) is real.
 
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One must understand 2 things under the following (admittedly simplifying) 2 assumptions:

i) The Hamiltonian is a compact self-adjoint operator on a complex separable Hilbert space.
ii) Its spectral equation is a PDE/ODE with real coefficients (one coefficient is the spectral value which is always real as the operator is s-adj).

1. A basis of the space of solutions of the PDE/ODE in ii) can always be chosen to be formed by real functions.
2. Even in the conditions of 1, generally, due to the fact that the Hilbert space where solutions of the spectral equations are sought is complex, a solution to the spectral equation is a complex function.
 
Boy oh boy. this is too complicated for my level!
 

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