# Homework Help: Help with simple bin. expan.

1. Apr 2, 2012

### perplexabot

1. The problem statement, all variables and given/known data
The allowed magnitudes of angular momentum are L = √(l(l+1)) * hbar. Use the binomial expansion to prove that when l is large, L ≈ (l + .5) * hbar.

2. Relevant equations
Binomial expansion formula: (1 + z)^n = 1 + nz + [n*(n-1)*z^2] / 2 + ....

3. The attempt at a solution

Ok. First I did:
L = √(l(l+1)) * hbar = [hbar * √l] * [1 + l]^.5
Then doing Bin. expan. apprx. for [1 + l]^.5,
I get a final answe, L ≈ hbar * √l * (1 + l/2)

Which is wrong. When compared to L ≈ (l + .5) * hbar, I can tell that I have an extra factor of √l, I think it has something to do with l being very large. Can someone please help me out? Thanks

Last edited: Apr 2, 2012
2. Apr 2, 2012

### Dick

Factor the l out of (l(l+1))^(1/2)=(l^2+l)^(1/2)=(l^2*(1+1/l))^(1/2).

3. Apr 2, 2012

### perplexabot

Thank you so much. I understand it. So:

[l * (l + 1)] ^ (1/2) = (l + l^2) ^ (1/2) = [l^2 * (l + 1/l)] ^ (1/2) = l * (1+ 1/l)^(1/2) ≈ l * (1 + 1/2l).

Great, that achieves the correct answer. Thanks again

Last edited: Apr 3, 2012
4. Apr 2, 2012

### Dick

In the last step it isn't equal, right? It's just an approximation. I'm also not sure where the minus signs came from and then disappeared to. Just a typo, right?

Last edited: Apr 3, 2012
5. Apr 3, 2012

### perplexabot

Yes, you are right, this is an approximation and that minus sign is a typo (I will edit my post). Thanks again.
One last optional question though, would you by any chance know why my first trial was wrong (my first post)?

6. Apr 3, 2012

### Dick

Sure. (1+x)^(1/2)~(1+x/2) is only a good approximation if x is small. So (1+1/l)^(1/2)~(1+1/(2l)) is a good approximation because 1/l is small. (1+l)^(1/2)~1+l/2 is a bad approximation because l is large.

7. Apr 3, 2012