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 -1	0	0 0 0	1	+1 0 -1
\downarrow \downarrow></math> (D-wave, 4%)	1	+1 +1 0 +1 0 -1 -1 0 +1	2	+2 +1 +1 0 0 0 -1 -1 -2	1	-1 0 -1 +1 0 -1 0 +1 +1