Rewriting a hydrogen state as a sum of eigenstates with negative coefficient

[Void123]

Homework Statement

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

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

Where $$R_{21}$$ is the radial equation.

I must rewrite $$\psi$$ in terms of summed eigenstates $$\psi_{nlm}$$.

Homework Equations

I assumed $$x = rsin\theta cos \varphi$$ and $$y = rcos\theta$$

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.

 

[nickjer]

You might need to show us your work, but imaginary normalization constants are perfectly reasonable. Since the only measurable quantity is:

$$\psi^{\dagger} \psi$$

Which shouldn't have any imaginary component in it.

Also, I think you mean to say:

$$z = r cos(\theta)$$