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

  • Thread starter Vitani11
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  • #1
Vitani11
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Homework Statement


Show that if H is a hermitian operator, then U = eiH is unitary.

Homework Equations


UU = I for a unitary matrix
A=A for hermitian operator
I = identity matrix

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...
 

Answers and Replies

  • #2
PeroK
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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...

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!
 
  • #3
Vitani11
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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?
 
  • #4
PeroK
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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.

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.
 
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  • #5
Vitani11
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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.
 
  • #6
Vitani11
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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.
 
  • #7
PeroK
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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.

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

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