Hermitian operator matrix

  • Thread starter Lolek2322
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  • #1
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Homework Statement


Eigenvalues of the Hamiltonian with corresponding energies are:
Iv1>=(I1>+I2>+I3>)/31/2 E1=α + 2β
Iv2>=(I1>-I3>) /21/2 E2=α-β
Iv3>= (2I2> - I1> I3>)/61/2 E3=α-β


Write the matrix of the Hamiltonian in the basis of the orthonormalized vectors I1>, I2>, I3>

If in t=0, system is in the state I1>, what is the wave function in t?

Homework Equations


Hij = <ilHlj>

The Attempt at a Solution


Although I know that energy is the eigenvalue of the Hermitian operator, I am not sure how to incorporate that in this certain problem. I have used mentioned equation for previous problems, but I always had the form of the operator. With only eigenvectors and eigenvalues I am stuck and don't even know how to begin solving this.
 
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Answers and Replies

  • #2
DrClaude
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There seems to be a part of the question that is missing. Can you write it out fully?
 
  • #3
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There seems to be a part of the question that is missing. Can you write it out fully?
I appologize. I have written it now
 
  • #4
DrClaude
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One way to go about this is to start by writing the Hamiltonian in the |v1>, |v2>, |v3> basis, then applying the proper transformation operation to "rotate" the Hamiltonian to the |1>, |2>, |3> basis.
 
  • #5
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But unfortunately I do not know how to do that
 
  • #6
kuruman
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The straightforward way to do it is
1. Find |1>, |2> and |3> as linear combinations of |v1>, |v2> and |v3> and verify that they are orthonormal.
2. Note that H|v1> = (α + 2β) |1> and get similar expressions for H operating on the other two v's.
3. Calculate things like < 1 | H | 2 > using the linear combinations from item 1 and substituting from item 2.
 

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