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

• Void123
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.

## 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.

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:

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

## 1. Can someone check my hypothesis and experiment design?

Yes, as a scientist, it is important to have your work peer-reviewed and validated by others in your field. You can ask a colleague or mentor to review your hypothesis and experiment design before conducting your experiment.

## 2. How do I know if my data collection and analysis is accurate?

To ensure the accuracy of your data collection and analysis, it is crucial to follow proper scientific methods and protocols. This includes using appropriate tools and techniques, collecting enough data points, and double-checking your calculations.

## 3. What should I do if my results are not what I expected?

If your results do not align with your hypothesis, it is important to consider possible sources of error and alternative explanations. It may also be helpful to consult with other scientists or conduct additional experiments to confirm your findings.

## 4. Can someone check my conclusions and implications?

Yes, it is important to have your conclusions and implications thoroughly reviewed before publishing your research. This can help identify any flaws in your reasoning and ensure that your conclusions are supported by your data.

## 5. How can I improve my scientific writing and presentation?

To improve your scientific writing and presentation, seek feedback from peers and mentors, read published research articles in your field, and attend workshops or courses on scientific communication. Practicing clear and concise writing and using visual aids can also enhance the effectiveness of your presentations.