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Group theory and quantum mechanics

  1. Apr 8, 2014 #1

    journeyman1986

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    Gold Member

    How to you get sets of complete basis functions using group theory ? For example , using triangle group for CH3 Cl ?
     
  2. jcsd
  3. Apr 8, 2014 #2

    dextercioby

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    Science Advisor
    Homework Helper

    The question's so general, you can write books about the answer. It has to do with the system's set of symmetries. If there's no symmetry, there's no group representation theory.
     
  4. Apr 10, 2014 #3

    journeyman1986

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    Gold Member

    Hi , I misunderstood the question sorry about that . I am new to group theory and quantum mechanics. Anyway,
    I did the problem and got the symmetrized basis functions .
    Next step would be to find the eigenvalues. The matrix I have is in block diagonal form.
    \left(
    \begin{array}{ccccc}
    \left(
    \begin{array}{c}
    \beta
    \end{array}
    \right) & \left(
    \begin{array}{c}
    -\delta
    \end{array}
    \right) & \left(
    \begin{array}{c}
    -\sqrt{3} \gamma
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) \\
    \left(
    \begin{array}{c}
    -\delta
    \end{array}
    \right) & \left(
    \begin{array}{c}
    \epsilon
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) \\
    \left(
    \begin{array}{c}
    -\sqrt{3} \gamma
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    -2 \alpha
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) \\
    \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    \alpha
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) \\
    \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    0
    \end{array}
    \right) & \left(
    \begin{array}{c}
    \alpha
    \end{array}
    \right)
    \end{array}
    \right)
    My question would be ? The basis functions from the 2-D representation looks like they are eigenfunctions but what about the 3*3 matrix that are in off diagonal form in the 5*5 matrix ? What do I do with that ?
     
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