# Hydrogen Wave Functions

## Homework Statement

The problem, along with a solution, is attached as an image file.

## The Attempt at a Solution

I have done the problem which was very straight forward. One simply had to look up the Rn,l and then plug in the appropriate quantum numbers. Since for a given n, there are n-1 values of l, there are two corresponding radial functions R2,0 and R2,1 for the n=2 state. So the probability density which is $$\left|R\right|^{2}$$ is the sum of the probability of being in the l=0 state and probability of being in the l=1 state. Because the problem does not indicate the state of the initial wave function, we don't know the coefficients of R2,0 and R2,1 so my TA writes that we should assume they are equally probable. But what I don't understand is why in his probability density function he writes a 1/4 in front of the l=0 function and a 3/4 in front of the l=1 function. Where did these values come from? I'm posting here instead of asking my TA because I have a midterm tomorrow morning and I won't get a response from my TA in time if I were to email him now.

Thanks

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fzero
There is 1 $$\ell=1$$ and 3 $$\ell=1$$ states. It appears that he gave each state weight 1/4.