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Show that if H is a hermitian operator, U is unitary

  1. Apr 18, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that if H is a hermitian operator, then U = eiH is unitary.

    2. Relevant equations
    UU = I for a unitary matrix
    A=A for hermitian operator
    I = identity matrix

    3. The attempt at a solution
    Here is what I have. U = eiH multiplying both by U gives UU = eiHU then replacing U with U-1 (a property of unitary matrices) I have UU = eiHU-1 and so UU = eiHe-iH = ei(H-H)=e0 = 1 = I. I don't think this is right though...
     
  2. jcsd
  3. Apr 18, 2017 #2

    PeroK

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    The bit I've underlined shows you assuming what you are supposed to prove.

    You need to show that ##UU^{\dagger} = I## without assuming it!
     
  4. Apr 18, 2017 #3
    Oh crap... okay I think I did it without that assumption. Here: U = eiH = ei(H†) = (eiH) = U therefore since U = U this proves it. What do you think?
     
  5. Apr 18, 2017 #4

    PeroK

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    That claims to prove that ##U = U^{\dagger}##, which is not what you are trying to prove. What you have is not right. You need to think more carefully about what ##(e^{iH})^{\dagger}## should be.
     
    Last edited: Apr 18, 2017
  6. Apr 18, 2017 #5
    Wait, is it even possible to go ahead and write that ei(H) is equal to (eiH)? because that is the same as treating the dagger symbol as some exponent. I will continue working on this and be back soon.
     
  7. Apr 18, 2017 #6
    U = eiH, U = (e-iH) = e-iH so multiplying the first equation by this on both sides yields UU = eiHe-iH = ei(H-H)=e0 = 1.
     
  8. Apr 18, 2017 #7

    PeroK

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    That looks better. Although, you should show why ##H## must be Hermitian.
     
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