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

1. Apr 18, 2017

### Vitani11

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. Apr 18, 2017

### PeroK

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. Apr 18, 2017

### Vitani11

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. Apr 18, 2017

### PeroK

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
5. Apr 18, 2017

### Vitani11

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. Apr 18, 2017

### Vitani11

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. Apr 18, 2017

### PeroK

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