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 ?