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

## 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...

PeroK
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2020 Award
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!

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?

PeroK
Homework Helper
Gold Member
2020 Award
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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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.

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.

PeroK