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Energy Spectrum of Two-State System

  1. Oct 5, 2011 #1
    1. The problem statement, all variables and given/known data

    A two-state system has Hamiltonian

    [itex]\sum |i\right\rangle hi \left\langle i| + Δ (| 1 \right\rangle \left\langle 2| + |2 \right\rangle \left\langle 1 |)[/itex]

    Where, [itex]\left\langle i | j \right\rangle = \deltaij[/itex], [itex]hi[/itex], and Δ are real.

    Compute the energy spectrum of this Hamiltonian.

    2. Relevant equations

    N/A

    3. The attempt at a solution

    What is this question asking me to do? What is meant by "energy spectrum"?

    Also; tried cleaning up the tex but something's not right and I can't seem to tell what (other than I'm on a different computer than I normally use).
     
    Last edited: Oct 5, 2011
  2. jcsd
  3. Oct 5, 2011 #2

    vela

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    The problem wants you to find all possible results if you measure the energy of the system.
     
  4. Oct 5, 2011 #3
    Thanks!
     
  5. Oct 5, 2011 #4
    Ok, my hamiltonian here is an hermitian operator plus a laplacian. This also tells me that |1> and <2| are vectors ([1,0] and [0,1] I think). As a painfully basic question of working with bras and kets, what is the operation (if that's the right word) |1><2| telling me to do?
     
  6. Oct 5, 2011 #5

    vela

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    Δ is just a number, not the Laplacian.

    Try calculating the matrix that represents the Hamiltonian in the [itex]\vert 1 \rangle[/itex] and [itex]\vert 2 \rangle[/itex] basis.
     
  7. Oct 5, 2011 #6
    Ah, ok, Δ being a number makes life a bit easier (I've just gotten use to it being a Laplacian every other time the prof uses it).

    I'm probably getting held up on notation (that I don't know what |1><2| means); and I'm not totally sure what you mean by calculating the matrix that represents the Hamiltonian in the [itex]\vert 1 \rangle[/itex] and [itex]\vert 2 \rangle[/itex] basis. Should this result in a diagonalized matrix?
     
  8. Oct 5, 2011 #7

    vela

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    You really need to go back and learn the basics of how operators and matrices are related. What I'm telling you to do is find the matrix
    \begin{bmatrix}
    \langle 1 | \hat{H} | 1 \rangle & \langle 1 | \hat{H} | 2 \rangle \\
    \langle 2 | \hat{H} | 1 \rangle & \langle 2 | \hat{H} | 2 \rangle
    \end{bmatrix}
    Surely your textbook goes over Dirac notation.
     
  9. Oct 5, 2011 #8
    I am, and it does (we're using both Merzbacher and Griffiths, leaves me in a bit of an information over load).
     
  10. Oct 6, 2011 #9

    vela

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    Let [itex]\hat{A} = \lvert a \rangle\langle b \rvert[/itex]. Say you want to calculate [itex]\langle \psi \lvert \hat{A} \rvert \phi \rangle[/itex]. You have
    [tex]\langle \psi \lvert \hat{A} \rvert \phi \rangle = \langle \psi \lvert (\lvert a \rangle\langle b \rvert) \rvert \phi \rangle[/tex]It works just like the notation suggests:
    [tex]\langle \psi \lvert \hat{A} \rvert \phi \rangle = \langle \psi \lvert \lvert a \rangle\langle b \rvert \rvert \phi \rangle = \langle \psi \vert a \rangle \langle b \vert \phi \rangle[/tex]
     
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