Thread: Finding a Complex Conjugate value of wave function.

First, sorry for my poor English and any impolite behavior might happen.

Here's two wave function(pic1) and problem below(pic2).
and they are polar coordinate problem ψ(r,θ,Φ)
You can see, problem requires conjugate function of ψ1.
Is it possible to find one? or is there a possibility that actually, problem requires complex conjugate function of ψ2 instead ψ1? (I mean, error of problem)

I have withdrew from school temporarily. so there's no one whom I can ask about this.
so if you can't answer it directly, please tell me how I can find matters about this subject.

Regards and sorry for my poor English again.

PS. There's no trouble integrating problem(pic2). So if you just confirm its possibility about complex conjugate value, I will appreciate you.

Originally Posted by boladore

First, sorry for my poor English and any impolite behavior might happen.

Here's two wave function(pic1) and problem below(pic2).
and they are polar coordinate problem ψ(r,θ,Φ)
You can see, problem requires conjugate function of ψ1.
Is it possible to find one? or is there a possibility that actually, problem requires complex conjugate function of ψ2 instead ψ1? (I mean, error of problem)

I have withdrew from school temporarily. so there's no one whom I can ask about this.
so if you can't answer it directly, please tell me how I can find matters about this subject.

Regards and sorry for my poor English again.

PS. There's no trouble integrating problem(pic2). So if you just confirm its possibility about complex conjugate value, I will appreciate you.

Assuming $\displaystyle a_0\,,\,r\in\mathbb{R}$ , and since we know that $\displaystyle \overline{e^{ir}}=e^{-ir}$, we get that:

$\displaystyle \overline{\Psi_1}=\frac{1}{2\sqrt{\pi}}\,\frac{1}{ a_0\sqrt{a_0}}\,2e^{r\slash a_0}$ , so $\displaystyle p_x=\int\overline{\Psi_1}qr\sin \theta \cos \phi \Psi_2 dV=-\int\frac{1}{2\sqrt{\pi}}\,\frac{1}{a_0\sqrt{a_0}} \,2e^{r\slash a_0}qr\sin \theta\cos \phi \,\frac{\sqrt{3}}{2\sqrt{2\pi}}\sin \theta e^{i\phi}$ $\displaystyle \frac{1}{2a_0\sqrt{2a_0}}\,\frac{re^{-r\slash 2a_0}}{\sqrt{3}a_0}=$

$\displaystyle \int\frac{1}{8\,a_0^4\,\pi}\,qr\sin \theta\sin \phi\, e^{i\phi}\,e^{r\slash 2a_0}\,dV$

Now, if either $\displaystyle a_0\,\,or\,\,r$ is a complex number then things get a little nastier (not that now that is a handsome expression, of course...)

Tonio

Thanks for your response.
But is it possible to get a Complex conjugate from exp(r/a0)? I mean, it has no imaginary part. Even if I think it as a coordinate system, it has only one variable "r".
So how can I get your first assuming?

I calculated it just with ψ1, and it lead proper value but not correct.
And I calculate it with C.C of ψ2 and its imaginary part vanished during intergrating. so there is no meaning of C.C

Regards.

Originally Posted by boladore

Thanks for your response.
But is it possible to get a Complex conjugate from exp(r/a0)? I mean, it has no imaginary part. Even if I think it as a coordinate system, it has only one variable "r".
So how can I get your first assuming?

I calculated it just with ψ1, and it lead proper value but not correct.
And I calculate it with C.C of ψ2 and its imaginary part vanished during intergrating. so there is no meaning of C.C

Regards.

That was a misreading of me: I thought it was $\displaystyle e^{ir\slash a_0}$, but there's no $\displaystyle i$ there , so assuming $\displaystyle a_o,r$ are real that exponential function is real and this it is its own conjugate...
So if you're asked to evaluate the conjugate of $\displaystyle \Psi_1$ then it equals to itsef, not so $\displaystyle \Psi_2$ which has $\displaystyle e^{i\phi}$ and thus perhaps that was a mistake in the question...

Tonio

Thanks. It seems that it has a error itself.