# Distance between electron and proton in Hydrogen

1. Dec 17, 2013

### skrat

1. The problem statement, all variables and given/known data
I had to calculate the probability that we find electron more than 0.1 nm away from proton in Hydrogen atoms if $\psi _{n,l,m}=\psi _{1,0,0}$ (i don't know the english word for this state, but I think we all know what we are talking about :D)
My result is 0.09957.

Now I am questioning myself if that makes any sense at all... Correct me if I am wrong: Bohr radius is expected value for distance between electron and proton in Hydrogen atoms for $\psi _{1,0,0}$.
If that is true, and if $r_B=5.29*10^{-11}m$ than my result makes some sense...

2. Relevant equations

3. The attempt at a solution

$\psi _{1,0,0}=R_{1,0}(r)Y_{0,0}(\theta ,\varphi )$ where $R_{1,0}=\frac{1}{\sqrt{4\pi }}\frac{2}{r^{3/2}}e^{-\frac{r}{r_B}}$

than $P(r>0.1 nm)=\int_{0.1 nm}^{\infty }\frac{1}{4\pi }\frac{4}{r^{3}}e^{-\frac{2r}{r_B}}4\pi r^2dr$

$P(r>0.1 nm)=\frac{1}{2}\int_{\frac{0.2 nm}{r_B}}^{\infty }u^2e^{-u} dr$

$P(r>0.1 nm)=\frac{1}{2}e^{\frac{0.2nm}{r_B}}(4(\frac{0.1 nm}{r_B})^{2}-4(\frac{0.1 nm}{r_B})+2)$

which should be 0.09957...

Does this sound ok?

2. Dec 17, 2013

### TSny

In your last expression, are you sure all of the signs are correct in the factor multiplying the exponential?

Also, is there a typo in the sign of the argument of the exponential?

3. Dec 17, 2013

### skrat

true, there is a mistake, let's say that $r_0=0.1 nm$ than

$P(r>r_0)=\frac{1}{2}\int_{\frac{2r_0}{r_B}}^{\infty }u^2e^{-u} dr$

this is now, according to a book: $\int x^2e^{ax}=e^{ax}(\frac{x^2}{a}-2\frac{x}{a^2}+2\frac{1}{a^3})$ in my case a=-1

so $P(r>r_0)=\frac{1}{2}(0-e^{-\frac{2r_0}{r_B}}(-\frac{4r_{0}^{2}}{r_{B}^{2}}-4\frac{r_{0}}{r_{B}}-2)$

which means that you are right...

$P(r>r_0)=\frac{1}{2} e^{-\frac{2r_0}{r_B}}(\frac{4r_{0}^{2}}{r_{B}^{2}}+4\frac{r_{0}}{r_{B}}+2)=0.272$

right?

4. Dec 17, 2013

### TSny

That looks good to me.

5. Dec 17, 2013

### skrat

Just one question:
Does bohr radius have any relation with the distance between the electron and proton?

6. Dec 17, 2013

### TSny

I'm not sure what you are asking. Can you be more specific?

7. Dec 17, 2013

### skrat

Is Bohr radius the average distance between the proton and electron or is it not?

In other words: what is bohr radius? I can't remember if my professor told us exactly what it is, so to me, at this moment, Bohr radius is just a number I don't really understand....

8. Dec 17, 2013

### TSny

Historically, the bohr radius is the radius of the circular orbit of the ground state in the old "Bohr model" of the hydrogen atom.

In the quantum mechanical description, the bohr radius happens to be the distance r from the proton at which the radial probability density is maximum.

9. Dec 17, 2013

### skrat

Thanks!

10. Dec 17, 2013