Difference between revisions of "Elong-14-04-16"

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(New page: =Deuteron Shape= From the [http://www.phy.anl.gov/theory/movie-run.html video make by S.C. Pieper, et al.], I extracted the tensor and vector polarization frames and made repeating videos...)
 
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=Deuteron Shape=
 
=Deuteron Shape=
  
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From the [http://www.phy.anl.gov/theory/movie-run.html video make by S.C. Pieper, et al.], I extracted the tensor and vector polarization frames and made repeating videos of each
+
From the [http://www.phy.anl.gov/theory/movie-run.html 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:
  
 
{|border="1"
 
{|border="1"
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|-
 
|-
 
| [[Image:2014-04-16-vector.gif]] || [[Image:2014-04-16-tensor.gif]]
 
| [[Image:2014-04-16-vector.gif]] || [[Image:2014-04-16-tensor.gif]]
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|-
 +
|}
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 +
=Deuteron States=
 +
 +
From basic quantum mechanics, we know that the possible states for 2 nucleons are
 +
 +
the isospin singlet with S=0:
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 +
:<math>|\uparrow \downarrow> - |\downarrow \uparrow></math> with <math>m_s=0</math>, <math>L=1</math>
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 +
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>
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 +
:<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>.
 +
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This leaves only two possible states:
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 +
:<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>
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 +
=Angular Momentum Analysis=
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Now, let's look at each of these a bit more indepth according to their angular momentum components
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{|border="1px" align="center" cellpadding="10" cellspacing="0"
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! State !! <math>J</math> !! <math>m_j</math>  !! <math>L</math> !! <math>m_l</math> !! <math>S</math> !! <math>m_s</math>
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|-
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| <math>|\uparrow \uparrow></math> (S-wave, 96%) || 1 || +1 || 0 || 0 || 1 || +1
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|-
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| <math>|\uparrow \uparrow></math> (S-wave, 96%) || 1 ||  0 || 0 || 0 || 1 ||  0
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|-
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| <math>|\uparrow \uparrow></math> (S-wave, 96%) || 1 || -1 || 0 || 0 || 1 || -1
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|-
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|
 +
|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 || +1 || 2 || +2 || 1 || -1
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 || +1 || 2 || +1 || 1 ||  0
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 ||  0 || 2 || +1 || 1 || -1
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 || +1 || 2 || 0  || 1 || +1
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 ||  0 || 2 || 0  || 1 ||  0
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 || -1 || 2 || 0  || 1 || -1
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 || -1 || 2 || -1 || 1 ||  0
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 ||  0 || 2 || -1 || 1 || +1
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|-
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| <math>|\downarrow \downarrow></math> (D-wave, 4%) || 1 || +1 || 2 || -2 || 1 || +1
 
|-
 
|-
 
|}
 
|}

Revision as of 16:26, 16 April 2014

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
2014-04-16-vector.gif 2014-04-16-tensor.gif

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
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