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Bound-state wave functions

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Why can bound-state wave functions be chosen to be real? The text states that, in one-dimensional problems, the spatial wave function for any allowed state can be chosen to be real-valued. Verify this using the following outline or some other method.

Please see the attached document for the questions, listed a) though c).

This question is from "An introduction to quantum physics" by French and Taylor.

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Solution Summary

We show why we can always choose any one-dimensional bounded-state wavefunction to be real-valued.

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