- #1
jono90one
- 28
- 0
Hi,
I am currently going over this and got me thinking about a scanario where you have A -> BC
Where A is S = 0, L=0, B is S = 1 L=0, C is S=1 L =0
(I'll use S = intrinsic spin, L = angular momentum, J = Total Angular momentum, |L-S|=< J =< L+S)
Maybe such a decay doesn't exist, but I'm just trying to test the "boundaries" of what I know about this topic.
For this reaction to happen the total angular moment must be J=0 (ie 2-2 = 0 as J = 0 on LHS so L=2 on RHS for total system)
[itex]\hat{P}|B> = (-1)(+1)(-1)^{L}[/itex]
L = 0
[itex]\hat{P}|B> = -1[/itex]
and like wise:
[itex]\hat{P}|C> = -1[/itex]
For charge conjugation
[itex]\hat{C}|B> = (-1)^{L+S}[/itex]
[itex]\hat{C}|B> = (-1)^{2} = +1[/itex]
[itex]\hat{C}|C> = (-1)^{2} = +1[/itex]
Now put them together:
[itex]\hat{P}|BC> = (-1)(-1)(-1)^{L}=(-1)^{2}= +1[/itex]
[itex]\hat{C}|BC> = (-1)(-1)(-1)^{L+S} = (-1)^{4} = +1[/itex]
Hence CP:
[itex]\hat{C}\hat{P}|BC> = (+1)(+1) = +1[/itex]
BUT
If we do the LHS using L = 0, S = 0 we get CP(A) = -1
I could be doing these calculations wrong, though this method has worked for easier examples (maybe by chance..). Or maybe it's not suppose to work ...?
[Ps this isn't homework, I'm just reading a book on it!]
Thanks for any help
I am currently going over this and got me thinking about a scanario where you have A -> BC
Where A is S = 0, L=0, B is S = 1 L=0, C is S=1 L =0
(I'll use S = intrinsic spin, L = angular momentum, J = Total Angular momentum, |L-S|=< J =< L+S)
Maybe such a decay doesn't exist, but I'm just trying to test the "boundaries" of what I know about this topic.
For this reaction to happen the total angular moment must be J=0 (ie 2-2 = 0 as J = 0 on LHS so L=2 on RHS for total system)
[itex]\hat{P}|B> = (-1)(+1)(-1)^{L}[/itex]
L = 0
[itex]\hat{P}|B> = -1[/itex]
and like wise:
[itex]\hat{P}|C> = -1[/itex]
For charge conjugation
[itex]\hat{C}|B> = (-1)^{L+S}[/itex]
[itex]\hat{C}|B> = (-1)^{2} = +1[/itex]
[itex]\hat{C}|C> = (-1)^{2} = +1[/itex]
Now put them together:
[itex]\hat{P}|BC> = (-1)(-1)(-1)^{L}=(-1)^{2}= +1[/itex]
[itex]\hat{C}|BC> = (-1)(-1)(-1)^{L+S} = (-1)^{4} = +1[/itex]
Hence CP:
[itex]\hat{C}\hat{P}|BC> = (+1)(+1) = +1[/itex]
BUT
If we do the LHS using L = 0, S = 0 we get CP(A) = -1
I could be doing these calculations wrong, though this method has worked for easier examples (maybe by chance..). Or maybe it's not suppose to work ...?
[Ps this isn't homework, I'm just reading a book on it!]
Thanks for any help