Hydrogen probability distribution

  • #1
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Main Question or Discussion Point

I have found that:

For l = 1:

[tex]\sum_{m=-l}^l |Y_l^m|^2 = \frac{3}{4\pi}[/tex]

For l = 2:

[tex]\sum_{m=-l}^l |Y_l^m|^2 = \frac{5}{4\pi}[/tex]

What significance does this have for the probability distribution in an hydrogen atom?
 

Answers and Replies

  • #2
jfizzix
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the significance is that there are five-thirds as many states of orbital angular momentum l=2, than for l=1. This makes sense because there are five orbitals for l=2, and three orbitals for l=1, for all the values of m.
 
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  • #3
Simon Bridge
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Do you know what those terms mean?
Do you know why you did the sum in each case?
 
  • #4
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Do you know what those terms mean?
Do you know why you did the sum in each case?
I think the numerator gives the degeneracy for a certain l?

since -l ≤ m ≤ l, m can take on 2l+1 possible values.

Like possible states for |n,l,m>:

For l = 1:

Possible states are |n,1,-1> and |n,1,0> and |n,1,1>
 
  • #5
Simon Bridge
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I think the numerator gives the degeneracy for a certain l?
Each spin/orbit state contributes a factor of ##1/4\pi## to the overall thingy being calculated.

But I meant the terms on the RHS.

For a particular |n,l,m> state, what does |Ylm|2 mean?

Now - what was your question?
 

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