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I am trying to use the k.p method to study quantum well band structure. One example Hamiltonian look like this [J. Appl. Phys., 116, 033709(2014)]

where

##{{\hat k}_ \pm } = {{\hat k}_x} \pm i{{\hat k}_y}##

and the matrix elements are function of ##{{\hat k}_i}##

and if quantum well is grown along z direction

the envelop functions have the form

\[F(x,y,z) = {e^{i{{\bf{k}}_\parallel } \cdot {\bf{r}}}}f(z)\]

and therefore

##\left\{ \begin{array}{l}

{{\hat k}_x} = - i\frac{\partial }{{\partial x}} \to {k_x}\\

{{\hat k}_y} = - i\frac{\partial }{{\partial y}} \to {k_y}\\

{{\hat k}_z} = - i\frac{\partial }{{\partial z}}

\end{array} \right.##

this Hamiltonian is an hermitian operator, but I am confused about this. For example, if I look at the (1,4) and (4,1) terms and use ${{\hat k}_z}=-i{\partial _z}$, the (1,4) term becomes

##-i{\partial _z}{P_1}##

and the (4,1) term becomes

##- i{P_1}{\partial _z}##

If I use finite difference method to turn these two terms into a matrix block, the factor -i is not going to flip sign and the matrix cannot he a hermitian matrix?

there must be something wrong with my understanding, please help.

##{{\hat k}_ \pm } = {{\hat k}_x} \pm i{{\hat k}_y}##

and the matrix elements are function of ##{{\hat k}_i}##

and if quantum well is grown along z direction

the envelop functions have the form

\[F(x,y,z) = {e^{i{{\bf{k}}_\parallel } \cdot {\bf{r}}}}f(z)\]

and therefore

##\left\{ \begin{array}{l}

{{\hat k}_x} = - i\frac{\partial }{{\partial x}} \to {k_x}\\

{{\hat k}_y} = - i\frac{\partial }{{\partial y}} \to {k_y}\\

{{\hat k}_z} = - i\frac{\partial }{{\partial z}}

\end{array} \right.##

this Hamiltonian is an hermitian operator, but I am confused about this. For example, if I look at the (1,4) and (4,1) terms and use ${{\hat k}_z}=-i{\partial _z}$, the (1,4) term becomes

##-i{\partial _z}{P_1}##

and the (4,1) term becomes

##- i{P_1}{\partial _z}##

If I use finite difference method to turn these two terms into a matrix block, the factor -i is not going to flip sign and the matrix cannot he a hermitian matrix?

there must be something wrong with my understanding, please help.