Chemical Forums
Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: Charkol on November 08, 2011, 09:51:34 PM
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I think I've been doing alright in pchem so far this semester. We are discussing term symbols, where you solve for S, L, and J
In my current problems, we are solving for all possible values for these. That means the clebsch-gordan series.
What the heck is this series?
The only form that I can find is:
S = s1 + s2 , s1 + s2 - 1 , ... , s1 - s2
This baffles me. Where does this negative 1 come in from, and what is hiding behind the ellipsis?
On wikipedia, there is a method which will give you only one solution, but I need 'all possible' solutions.
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For example, let s1 = 2 and s2 = 2.
S, then would be s1 + s2 , s1 + s2 - 1 , ... , s1 - s2
or 4, 3, 2, 1, and 0
Thats s1+s2, s1+s2-1, s1+s2-2, s1+s2 -3, and so on until you get to s1-s2.
If s1 = 3 and s2 = 2, S would be 5,4,3,2, and 1.
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That was how understood it, but when you come to the situation where, maybe
s1 = 2 and s2 = 0...
You would have
S = 2 + 0, 2 + 0 - 1,..., |2 - 0|
You would think that you would have the values of
2, 1, 0
But this is not true.
What S values you can have here are:
just 2
Because you can have values between the max and min values of the S series:
max = s1 + s2
min = |s1 - s2|
Those values would be between 2 and 2, which is just one value of 2.
Of course, s1 and s2 are typically referred to for the spin values of the electrons either (+1/2) or (-1/2).
But this example does hold for orbital angular momentum, l, values, but this is a confusing symbol to use in this typing.
Thanks for the help, this isn't very difficult, but I had a monster of a mental block while trying to figure this out.
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That was how understood it, but when you come to the situation where, maybe
s1 = 2 and s2 = 0...
You would have
S = 2 + 0, 2 + 0 - 1,..., |2 - 0|
You would think that you would have the values of
2, 1, 0
But this is not true.
What S values you can have here are:
just 2
Because you can have values between the max and min values of the S series:
max = s1 + s2
min = |s1 - s2|
Those values would be between 2 and 2, which is just one value of 2.
Of course, s1 and s2 are typically referred to for the spin values of the electrons either (+1/2) or (-1/2).
But this example does hold for orbital angular momentum, l, values, but this is a confusing symbol to use in this typing.
Thanks for the help, this isn't very difficult, but I had a monster of a mental block while trying to figure this out.
This is a mathematical question more than chemical.
The series in the OP gives all the acceptable values between (s1 + s2) and (s1 - s2). In your case for s1 = 2 and s2 = 0, this means all the acceptable values between 2+0 and 2-0, that is the only possible value is 2.
That is like giving the series R(N) = 1 + 2 + ยทยทยท + N. For R(3) = 6, but R(2) = 3 not 5 and R(1) = 1 not 4.
It is a question of ambiguity on the mathematical notation. Rigorously you would say that the definition for R(N) is valid when N>=3 and then give definitions for R(2) and R(1) or you can interpret the series correctly and get the correct values.
The same about the series in the OP. You can interpret it correctly as all the acceptable values between (s1 + s2) and (s1 - s2) or you can write down a more complex mathematical and unambiguous expression {*}.
{*} You will need the latter option if you are programming a computer.