Elong-14-04-16
Deuteron Shape
From the video make by S.C. Pieper, et al., I extracted the tensor and vector polarization frames and made repeating videos of each. When we vector-polarize or tensor-polarize, the probability densities for the deuteron look like:
| Vector | Tensor |
|---|---|
 |
|
Tensor Polarization Relation to Vector Polarization
Tensor polarization is related to the vector polarization by <math>P_z=2-\sqrt{4-3P_{zz}^2}</math>
Deuteron States
From basic quantum mechanics, we know that the possible states for 2 nucleons are
the isospin singlet with S=0:
- <math>|\uparrow \downarrow> - |\downarrow \uparrow></math> with <math>m_s=0</math>, <math>L=1</math>
and the isospin triplet with S=1:
- <math>|\uparrow \uparrow></math> with <math>m_s=+1</math>, <math>L=0</math>
- <math>|\uparrow \downarrow> + |\downarrow \uparrow></math> with <math>m_s=0</math>, <math>L=1</math>
- <math>|\downarrow \downarrow></math> with <math>m_s=-1</math>, <math>L=2</math>
For the deuteron, <math>J=1</math> and <math>P=+1</math>. This kills both of the <math>m_s=0</math> states, since they cannot simultaneously have <math>J=1</math> and <math>P=+1</math> since <math>P=(-1)^L</math>.
This leaves only two possible states:
- <math>|\uparrow \uparrow></math> with <math>m_s=+1</math>, <math>L=0</math>
- <math>|\downarrow \downarrow></math> with <math>m_s=-1</math>, <math>L=2</math>
Angular Momentum Analysis
Now, let's look at each of these a bit more indepth according to their angular momentum components
| State | <math>J</math> | <math>m_j</math> | <math>L</math> | <math>m_l</math> | <math>S</math> | <math>m_s</math> |
|---|---|---|---|---|---|---|
| \uparrow \uparrow></math> (S-wave, 96%) | 1 | +1 | 0 | 0 | 1 | +1 |
| \uparrow \uparrow></math> (S-wave, 96%) | 1 | 0 | 0 | 0 | 1 | 0 |
| \uparrow \uparrow></math> (S-wave, 96%) | 1 | -1 | 0 | 0 | 1 | -1 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | +1 | 2 | +2 | 1 | -1 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | +1 | 2 | +1 | 1 | 0 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | 0 | 2 | +1 | 1 | -1 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | +1 | 2 | 0 | 1 | +1 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | 0 | 2 | 0 | 1 | 0 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | -1 | 2 | 0 | 1 | -1 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | -1 | 2 | -1 | 1 | 0 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | 0 | 2 | -1 | 1 | +1 |
| \downarrow \downarrow></math> (D-wave, 4%) | 1 | +1 | 2 | -2 | 1 | +1 |
