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Propagator with Pauli matrices

  1. Jul 9, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a 1/2-spin particle. Its time evolution is ruled by operator [itex]U(t)=e^{-i\Omega
    t}[/itex] with [itex]\Omega=A({\vec{\sigma}}\cdot {\vec{L}})^{2}[/itex]. A is a constant. If the state at t=0 is described by quantum number of [itex]{\vec{L}}^2[/itex], [itex]L_{z}[/itex] and [itex]S_{z}[/itex], [itex]l=0[/itex], [itex]m=0[/itex] and [itex]s_{z}={1/2}[/itex], determinate the state at a generic time t.
    2. Relevant equations

    [itex]({\vec{\sigma}}\cdot {\vec{L}})^{2}={{\vec{L}}^{2}}-{{\vec{\sigma}}\cdot{\vec{L}}}[/itex]
    and
    [itex](\vec{\sigma}\cdot\vec{u})^{2n}=1[/itex] with [itex]2n[/itex] even number and [itex]\vec{u}[/itex] a unitary vector

    3. The attempt at a solution
    I've used the relations I've written above to write the propagator as [itex]e^{-iA{\vec{L}}^{2}t}[/itex][itex]e^{iA{{{\vec{\sigma}}\cdot{\vec{L}}}}t}[/itex] and I've found out

    [itex]e^{-iA{\vec{L}}^{2}t}[{cos(At)+i{({\vec{\sigma}}\cdot\vec{L})}sin(At)]}[/itex]. But I don't think it is correct because L is not a versor.

    Thanks
     
  2. jcsd
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