Finding particle probability without wave function given

[Saptarshi Sarkar]

Homework Statement

If the operator $\hat A = i(\hat x^2 + 1)\frac d {dx} + i\hat x$ is Hermitian, then calculate the probability of finding a particle (satisfying the condition $\hat Aψ(x) =0$) in the region -1<x<1.

Relevant Equations

P = $\int_{-1}^1 {|ψ|}^2 \, dx$

I know how to calculate the probability of finding the particle in a region by integrating the mod square of the wave function within that region. But in this question only the operator is provided but not the wave function. I am not sure how am I supposed to proceed with this problem.

 

[vela]

An explicit wave function wasn't given, but one was implicitly provided. It's the one that satisfies $\hat A \psi = 0$. But first you need to determine if $\hat A$ is Hermitian.

 

[PeroK]

I must admit I'm struggling to make sense of that condition. $\hat A$ doesn't look Hermitian. And how would that affect the differential equation for $\psi$?