Understanding Probability Densities for Hydrogen Wave Functions

[CanIExplore]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I have done the problem which was very straight forward. One simply had to look up the R_n,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 R_2,0 and R_2,1 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 R_2,0 and R_2,1 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

 

[fzero]

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

 

[CanIExplore]

You mean 1 l=0 state, but yes I see now. This makes perfect sense. Thank you very much!