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Hi! I'm going through this post now, adding explanations of terms that are undefined, and fixing grammar and formatting:
A question: is it really wise to use $\psi_+$ as the name for the state whose energy is $E_0 - \Delta$, and $\psi_-$ as the name for the state whose energy is $E_0 + \Delta$? I guess I know why Piotr is doing this, but it could be a bit confusing. Maybe I'll add an explanation.
By the way, the term 'superposition' is never defined. I'll also add a definition of that, since that's the topic of the post!