Calculating Normalization Factor for 1s Atomic Orbital of H

[geronimo123]

Hi all! I hope somebody is able to help me on my way with this question.

I have been asked to show that the Normalization factor for the 1s atomic orbital of H is $$1/(\Pi a_o^3)^\frac{1}{2}$$. The wavefunction is $$\psi(r) = N exp(-r / a_o)$$

I'm given $$dt = r^2 sin \Theta$$ and $$dr d\Theta d\Phi$$ and $$\int_{0}^{\infty}x^n e^{ax} dx=n!/a^n+1$$

I must admit I'm clueless which direction to go. It was mentioned to me, that squaring the wavefunction is the first step, but I cannot arrive at the given constant. Am I starting off on the wrong foot?

Thanks for any input, in advance.

geronimo

 

[nrqed]

Hi all! I hope somebody is able to help me on my way with this question.

I have been asked to show that the Normalization factor for the 1s atomic orbital of H is $$1/(\xi a_o^3)^\frac{1}{2}$$. The wavefunction is $$\psi(r) = N exp(-r / a_o)$$

I'm given [tex]d \Tau = r^2 sin \Theta[tex];$$dr d\Theta d\Phi$$ and $$\int_{0}^{\infty}x^n\e^-ax\dx=n!/a^n+1$$

I must admit I'm clueless which direction to go. It was mentioned to me, that squaring the wavefunction is the first step, but I cannot arrive at the given constant. Am I starting off on the wrong foot?

Thanks for any input, in advance.

geronimo

You have all the pieces in place.
Just impose $\int_0^\infty dr r^2 \int_0^{2 \pi} d\phi \int_0^\pi sin(\theta) d\theta \,\,N^2 e^{-2 r /a_0} =1$ and solve for N.

 

[geronimo123]

I must admit that my integration is not what it used to be. I'm taking some physical chemistry night classes where this problem was set. How might one initially solve for N?

 

[freeman]

I must admit that my integration is not what it used to be. I'm taking some physical chemistry night classes where this problem was set. How might one initially solve for N?

The $$\phi$$ integral just adds a factor of $$2\pi$$, change variables in the $$\theta$$ integral to $$x=\cos(\theta)$$ and then it becomes a lot nicer.

Good luck.
Eoin Kerrane.

 

[nrqed]

I must admit that my integration is not what it used to be. I'm taking some physical chemistry night classes where this problem was set. How might one initially solve for N?

the theta integral gives 2. The phi integral gives 2 pi. For the r integral, you have a formula. just compare the r integral to the formual you have, term by term (what is n? etc) and you will have the result of the r integration. Then you have N^2 times an expression = 1 and you solve for N

 

[geronimo123]

Super guys, I have managed it finally thanks to your help. Best regards