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Matrix of Hamiltonian, system's state - quantum

  1. Dec 8, 2013 #1
    Hello
    I'm reaaaaaally stuck here, if someone can explain me it'd be great because i just dn't know how to do all that

    1. The problem statement, all variables and given/known data
    Consider a system with the moment of inertia l=1
    A base of the space of the states is constituted by the three eigenvectors of Lz |+1> , |0>, |-1> of eigenvalues +h, 0, -h (h=h/2π)
    The Hamiltonian has the form : H=wo/h(Lu²-Lv²)

    where Lu and Lv are the component of L in the directions Ou & Ov of the xOz plan, 45° from Ox and Oz. wo is a real constant.

    a. Write the matrix representing H in the basis |+1> , |0> and |-1>
    What are the stationnary states of the system and their energies ?
    b. At t=0 the system is in the state

    |ψ(0)>=1/√2(|+1>-|-1>)
    What is |ψ(t)> at time t ? If we measure Lz what are the probabilities of the different results ?


    c. Calculate <Lx>(t) <Ly>(t) & <Lz>(t)



    2. Relevant equations



    3. The attempt at a solution
    a.I am soooo lost I know I have to find Lx and Lz first , then Lu & Lv to find H
    But I just don't know how to do that

    b. We have
    |ψ(t)>=|ψ(0)>exp(-iHt/h)
    |+1> and |-1> are not eigenstates of H so I have to express them in eigenstates of H but how ?
    And I will have to convert the energies |En> found in term of |+1> AND |-1> , how ?

    c.

    Thanks
     
  2. jcsd
  3. Dec 8, 2013 #2

    Simon Bridge

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    ##\renewcommand{\br}[1]{| #1 \rangle} \renewcommand{\kt}[1]{\langle #1 |}##
    Lets see if I can translate:
    System with moment of inertia ##I=1## in some units.
    Use a basis of eigenvectors of z-angular-momentum ##\{ \br{+1}, \br{0}, \br{-1} \}##

    ##L_z\br{+1}=\hbar\br{+1},\; L_z\br{0}=0,\; L_z\br{-1}=-\hbar\br{-1}##

    The Hamiltonian has form: $$\hat{H}=\frac{\omega_0}{\hbar}\left( L_u^2-L_v^2 \right)$$

    The u and v directions are what??
    ##L_z## is the component of ##\hat{L}## in the z direction ...

    Yes you need to know how the u and v components relate to the x and z components. You were told - but I'm not sure I follow your notation so you'll have to go back to your notes to see what it means.

    But they are eigenstates of ##L_z##
    You need to be able to expand one set of eigenstates in terms of the others - it's a change of basis.

    In the previous you found the eigenstates of ##H## in terms of those of ##L_z## ... so you need to do the first one, well, first.
     
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