3bChannels
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Three-body states:
General remarks:
- Three-body states are partially antisymmetric; i.e., they are antisymmetric in the exchange of particles 1 and 2, but not with respect to the third particle.
- Three-body states are built upon two-body channels {$ \alpha_2 $}, that are antisymmetric and have q.n. {$ l, s, j, t $}. See 2bChannels
- Triangular condition for the total spin J: {$ |I_3 - j| \le J \le |I_3 + j| $}.
- Total parity given by {$ \Pi = (-1)^{l+\lambda} $}.
- Triangular condition for the total isospin T: {$ |1/2 - t| \le T \le |1/2 + t| $}.
Three-body channels are constructed in the following order:
- Start with the first two-body state from the list generated according to 2bChannels
- A two-body state with fixed j will imply a list of possible spins {$ I_3 $} of the third particle.
- Channels are created by increasing {$ I_3 $}.
- For each {$ I_3 $}, the total parity determines the single allowed value of {$ \lambda $}.
- Continue to the next two-body channel and goto step 2.
Explicit example for the most important triton channel.
Consider the 1S0 channel in which j=0. Which lambda's can we have that couple to total J=1/2
We have to fulfill the triangle relation {$ |I-j|\leq 1/2 \leq I+j $}
First consider {$1/2\leq I+j$} thus {$I \geq 1/2-j$} which is always fulfilled since I is at least 1/2
Then consider {$|I-j| \leq 1/2 $}
{$I>j \longrightarrow j<I \leq 1/2+j $}
{$ I<j \longrightarrow j-I \leq 1/2 \longrightarrow j-1/2 \leq I<j $}
{$j-1/2 \leq I \leq j+1/2$}
for j>0 (Note that for j=0 there will be only one lambda) we have always 2 I. Now the question is how many lambdas we have for each I.
Note that I is at least 1/2
{$|\lambda-1/2| \leq l \leq \lambda+1/2$}
rhs
{$ \lambda \geq I-1/2 $}
The left-hand-side gives
{$\lambda >1/2 \longrightarrow I-1/2 \leq \lambda \leq I+1/2$}
{$ \lambda<1/2 \longrightarrow 1/2-I \leq \lambda$}
That means we have maximally 4 channels The triton has even parity so we only need to consider those states where (-1)^l+lambda=+1. Since the allowed lambdas differ by 1, only one of them is allowed. That means we have maximally 2 channels attached to each l