- #1
fluidistic
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Homework Statement
The normalized wave-function corresponding to the ground state hydrogen atom has the form [tex]\Psi _0 = \frac{1}{\sqrt \pi} \cdot \frac{e^{-r/a_0}}{a_0 ^{3/2}}[/tex] where [tex]a_0[/tex] is Bohr's radius.
1)Graph the density of probability to find the electron in a spherical edge between r and r+dr. (Sorry "spherical edge" might not be the words but I hope you can understand. If not, let me know).
2)Calculate the probability that an electron in this state can be found at a greater distance than [tex]a_0[/tex].
3)Calculate the most probable distance between the electron and the nucleus.
4)Calculate [tex]\langle r \rangle[/tex].
Homework Equations
[tex]\int _0 ^{\infty} \Psi _0 ^2 dr =1[/tex]. I omit the absolute value since the function seems positive on all its interval of definition.
The Attempt at a Solution
For 2), my intuition tells me "[tex]1-\int _0 ^{a_0} \Psi _0 ^2 dr[/tex]" or "[tex]\int _{a_0} ^ \infty \Psi _0 ^2 dr[/tex]".
I'm having a hard time with 1).
I'm trying to find out the probability to find the electron between any radius of length dr. By any radius I mean a radius of length dr that is far away from the nucleus by an arbitrary number.
I've done an attempt on it, lead to nothing. I called [tex]C=\frac{1}{\sqrt \pi a_0 ^{3/2}}[/tex] and so what I'm looking for should be of the form [tex]C^2 \int _r^{r+dr}e^{-\frac{2r}{a_0}} dr[/tex]. I must be confusing variables calling all r's... Anyway I could never get rid of "dr" and I know my result must depend on r and not on dr.
Any insight is appreciated.