How to Rewrite a Hydrogen State in Terms of Summed Eigenstates | Homework Help

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In summary, the conversation is regarding finding a wave function for a hydrogen state and rewriting it in terms of summed eigenstates. The attempt at a solution involved finding four different eigenstates, but one of them had a (-1) coefficient leading to an imaginary normalization constant. The poster is unsure of any mistakes made and there is a discussion about the measurable quantity and clarification on the value of z.
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Homework Statement



I am giving the following wave function which describes a hydrogen state:

[tex]\psi(r, 0) = (\frac{A}{\sqrt{\pi}})(\frac{1}{a_{0}})^{3/2} exp(-\frac{r}{a_{0}}) + (1/\sqrt{12*\pi})(\frac{z - \sqrt(2)x}{r})R_{21}[/tex]

Where [tex]R_{21}[/tex] is the radial equation.

I must rewrite [tex]\psi[/tex] in terms of summed eigenstates [tex]\psi_{nlm}[/tex].



Homework Equations



I assumed [tex]x = rsin\theta cos \varphi[/tex] and [tex]y = rcos\theta[/tex]



The Attempt at a Solution



I come up with four different eigenstates, but one of them has a (-1) coefficient (which leads to an imaginary normalization constant, A).

I don't see what I could have done wrong though.
 
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  • #2
You might need to show us your work, but imaginary normalization constants are perfectly reasonable. Since the only measurable quantity is:

[tex]\psi^{\dagger} \psi[/tex]

Which shouldn't have any imaginary component in it.

Also, I think you mean to say:

[tex]z = r cos(\theta)[/tex]
 

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