Dirac equation continuity issue

[raymo39]

So I definitely believe that the continuity of the Dirac equation holds, there is one thing that annoys me, which is that

$$c \alpha . (-i \hbar \nabla \psi ) = c (i \hbar \nabla \psi^\dagger ) . \alpha$$

from the first part of the Dirac Hamiltonian
because the momentum operator should be Hermitian, but there is clearly a change of sign. I realize that the Hermitian conjugate of the spinor is now differently oriented, but shouldn't there be a change in ∇ such that its hermitian adjoint is negative itself to preserve the Hermitian nature of p?
I'm quite happy to try and prove this myself, but I am a bit lost as to where to start

 

[andrien]

momentum operator is defined as in Schrödinger one,i.e. p=-ih^-∇,which is hermitian as everyone knows.