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Exponential of hermitian matrix

  1. Feb 12, 2017 #1
    1. The problem statement, all variables and given/known data

    Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential.

    a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are exp[-i*a subscript(i)].

    b) Show that U is unitary.

    This example is important in quantum mechanics when A=Ht/h, where H is the Hamiltonian operator, t is time, and h is Plank's constant. Then U evolves the wavefunction over a time t.
    2. Relevant equations
    I have uploaded what I have come up with so far, but not sure where to go with it

    3. The attempt at a solution

    See uploaded file
     

    Attached Files:

  2. jcsd
  3. Feb 12, 2017 #2

    blue_leaf77

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    You have ##A\nu_i = a_i \nu_i## where ##a_i## is the eigenvalue corresponding to an eigenvector ##\nu_i##. Have you tried inputting this eigenvalue equation into the eigenvalue equation for ##U## using its Taylor expansion definition?
    Do you know the property of the determinant of a unitary matrix?
     
  4. Feb 12, 2017 #3
    Ok, will see if I can make that work. Thanks!
     
  5. Feb 12, 2017 #4
    I'm trying to figure this out, but not sure how to input that equation into the Taylor expansion
     
  6. Feb 12, 2017 #5

    blue_leaf77

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    Try applying ##U## in its Taylor form to an eigenvector of ##A##
    $$
    (I + iA - \frac{1}{2}A^2 - i\frac{1}{6} A^3 + \ldots) \nu_i .
    $$
     
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