# Make an operator to be hermitian

1. Jul 16, 2014

### masoodsa

I have an operator which isnot Hermitian is there any way to make it hermitian ?

2. Jul 16, 2014

### ChrisVer

add its hermitian conjugate and redefine it.

3. Jul 16, 2014

### Staff: Mentor

Any operator can be written of the form H = A + iB where A and B are Hermitian.

In fact one could base QM on any operator using that result via the obvious extension to the Born Rule E(H) = E(A) + i E(B)

Interestingly in proving the Born rule via Gleason that's what one does.

So why does one stick with Hermitan operators? The reason is the properties of continous transformations between states and symmetry. Wigners Theroem implies such must be Unitary:
http://en.wikipedia.org/wiki/Wigner's_theorem

If U is such a transformation dependant on t (not necessarily time) and we consider Δt small. U(Δt) = 1 + D Δt. UU(bar) = 1 + DΔt + D(bar)Δt = 1 or D = -D(bar) which means D is pure imaginary. If we want to associate D with an observation, which from symmetry considerations you want to do because it leads to the same equations as classical physics (see Ballentine chapter 3), then its expectation would be imaginary as well which would be a rather strange thing. So what we do is define divide D by i and use that operator which is of course real.

Thanks
Bill