# Calculating the probability of finding an electron

## Homework Statement

The probability of finding a 1s electron in a region between r and r+dr is:

probability = $$(4/a^3_o) r^2 e^{-2r/a_o} dr$$

1. work out the probability that an electron would be found in a sphere of radius $$a_o$$

## Homework Equations

I know to find the probabilty you work out $$\int|\psi|^2 dr$$ but because the probability is already given what do i do?

The probability is given in a region between r and r+dr so i guess I somehow work it out over 360 degress?

Thanks

Last edited:

Simon Bridge
Homework Helper
You have been provided with the probability density function - you need the probability between some limits... how would you normally do that? Think back to probability and statistics work you did in math class.

You have been provided with the probability density function - you need the probability between some limits... how would you normally do that? Think back to probability and statistics work you did in math class.

are the limits ao and 0 and then intergrate by parts?

Okay so if what i said is true:

I changed the variable to make it more straight forward (x=r/ao) so
(upper limit is 1 lower is 0)

probability = $$(4/a_o) x^2 e^{-2} dx$$
Then using intergration by parts I get

$$(4/a_o) \int x^2 e^{-2x} dx =$$
$$= 4/a_o [ (-(1/2)x^2 e^{-2x} - \int (-1/2)2x e^{-2x} dx) ]$$
$$= 4/a_o [(-(1/2)x^2 e^{-2x} - (1/2) x e^{-2x} - \int -(1/2) e^{-2x} dx)]$$
$$=4/a_o [(-(1/2)x^2 e^{-2x} - (1/2) x e^{-2x} - (1/4) e^{-2x} )]$$

so substituting in the upper and lower limits (1 and 0) i get

$$=(1/a_o) -(5e^{-2}/a_o)$$

If someone could check this for me or tell me where I have gone wrong. The problem i find is that the probability is a 11 digit number

Last edited:
Simon Bridge
Homework Helper
Why is an 11 digit answer a problem?

Why is an 11 digit answer a problem?

Because the wavefunction is normalized it should be a maximum of 1 :). Ive solved it now anyway, this website helped confirm my answer if anyone has a similar problem

Code:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydrng.html#c1

Simon Bridge
Homework Helper
Cool, well done.

Just some general notes:

You should be getting used to reasoning out your answers rather than relying on some outside authority. That's why I was being cautious about saying "yep - that's how you do it". How do you know I'm right? How do you know that website is right - maybe someone made a mistake?

Scientists can come across as very arrogant but one of the humilities in science is this distrust of argument by authority. A Nobel-Prize-winner can be challenged on the same grounds as anyone else.

0.12345678901 is an 11 digit number less than 1. See why I asked why an 11 digit answer was a problem?

You can make hyperlinks by putting the urls in tags. If you just paste the link i...phy-astr.gsu.edu/hbase/quantum/hydrng.html#c1

But you can also manually type them in for tidier links like this.
(Assuming you are using the quick reply box.)