Hydrogen atom (finding electron probability)

In summary: The integral can be simplified by expanding the term and multiplying by e^(-r/a), then I have integral of r^4*e^(-r/a)I'll just use e^{\frac{-r}{a}} \approx 1-\frac{r}{a}. r_{m} is small enough for me.thanks.
  • #1
natugnaro
64
1
[SOLVED] Hydrogen atom (finding electron probability)

Homework Statement


For electron in eigensate of Hydrogen we have these expectation values

<r>= 6a , <r^-1>= 1/4a

(a is Bohr radius)

a) find that eigenstate.
b) find the probability of finding electron in region 0 < phi < Pi/6 , 0 < theta < Pi/8
and 0 < r < rm
where rm is the first minimum of radial wavefunction in which electron is.



Homework Equations





The Attempt at a Solution



a) Using <nl|r^-1|nl>=1/((n^2)*a) it follows that n=2

then using <nl|r|nl>=1/2*[3*n^2-l(l+1)]a I got l=0 ,
so the eigenstate must be Psi(200).

b) To find the probability, I have to integrate

[tex]P=\int|\psi_{200}|^{2}r^{2}Sin\theta drd\phi d\theta = \frac{1-Cos(\frac{\pi}{8})}{(2a)^{3}6}\int^{r_{m}}_{0}(1-\frac{r}{2a})^{2}e^{\frac{-r}{a}}r^{2}dr[/tex]

I can do the integral using tables, but can this integral be simplified. I only have to integrate from 0 to rm so if rm is much less than Bohr radius a, I can make an approximation (series expansion of e^x).
I'm not shure is rm<<a ? , also how are rm and a (Bohr radius) related ?
 
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  • #2
How do you determine "r_{m}" ? From what equation ?
 
  • #3
[tex]\frac{\partial}{\partial r}(R_{20}(r))=0 \Rightarrow \frac{e^{-r/2a}(-4a+r)}{4\sqrt{2}a^{7/2}}=0 \Rightarrow r_{m}=4a[/tex]

So, rm is four times smaller than a.
Well, I think this is ok, since I'm just trynig to avoid some integral, not build something that suppose to work :smile: .
 
  • #4
You can also expand the [tex](1-\frac{r}{2a})^{2}[/tex] term, multiply with [tex]r^2[/tex] and integrate by parts. Quite painful, but exact!
 
  • #5
PhysiSmo said:
You can also expand the [tex](1-\frac{r}{2a})^{2}[/tex] term, multiply with [tex]r^2[/tex] and integrate by parts. Quite painful, but exact!

and multiply by e^(-r/a), then I have integral of r^4*e^(-r/a)
 
  • #6
I'll just use [tex]e^{\frac{-r}{a}} \approx 1-\frac{r}{a}[/tex]. [tex]r_{m}[/tex] is small enough for me.
thanks.
 
  • #7
I meant that you could integrate by parts the integral

[tex]\int_0^{r_m} r^4e^{-r/a}dr=r^4\frac{e^{-r/a}}{-1/a}\left|_0^{r_m}-\int_0^{r_m} 4r^3e^{-r/a}\left(-\frac{1}{a}\right)dr=\cdots[/tex]

and so on...
 
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  • #8
ah, yes , tahnks again.
 

Related to Hydrogen atom (finding electron probability)

1. What is the electron probability in a hydrogen atom?

The electron probability in a hydrogen atom refers to the likelihood of finding an electron at a specific location around the nucleus. This probability is represented by a mathematical function called the wave function, which is used to calculate the likelihood of an electron being at a certain distance from the nucleus.

2. How is the electron probability distribution calculated in a hydrogen atom?

The electron probability distribution in a hydrogen atom is calculated using the Schrödinger equation, which is a mathematical equation that describes the behavior of quantum particles, such as electrons. This equation takes into account the properties of the electron, the properties of the nucleus, and the distance between them to determine the probability of finding the electron at a specific location.

3. What is the significance of the electron probability in a hydrogen atom?

The electron probability in a hydrogen atom is significant because it helps us understand the behavior and properties of atoms. By knowing the likelihood of finding an electron at a certain location, we can determine the electron's energy level, orbital shape, and other characteristics that are crucial in understanding chemical bonding and reactions.

4. How does the electron probability change in different energy levels of a hydrogen atom?

The electron probability changes in different energy levels of a hydrogen atom due to the different shapes and sizes of the orbitals. As the energy level increases, the orbitals become larger and more complex, resulting in a higher probability of finding the electron further away from the nucleus.

5. Can the electron probability be visualized?

While the electron probability cannot be directly visualized, it can be represented graphically through electron density plots or probability density maps. These visualizations show the relative probability of finding an electron at different locations around the nucleus, providing a better understanding of the electron's behavior in a hydrogen atom.

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