Is the Hermitian Conjugate of an Operator Always Hermitian?

[danny271828]

Homework Statement

a = x + \frac{d}{dx}

Construct the Hermitian conjugate of a. Is a Hermitian?

2. The attempt at a solution

<\phi|(x+\frac{d}{dx})\Psi>

\int\phi^{*}(x\Psi)dx + <-\frac{d}{dx}\phi|\Psi>

I figured out the second term already but need help with first term... am I on the right track?

 

[Dick]

Well, x is real. It's the position operator. So x^*=x.

 

[dextercioby]

HINTS:

1.What's the domain of "a" as an operator in the L^{2}(\mathbb{R},dx) ?
2. Stick to that domain. Consider the matrix element of that operator among 2 vectors in that Hilbert space. What restrictions do you get when trying to find the adjoint ? Therefore ?
3. Does the adjoint exist ?
4. What's its domain ?
5. Is the "a" operator hermitean/symmetric ?