The crystal momentum $p_1 =\hbar\cdot k $ and this is defined in the reciprocal momentum, I am guessing this $p_1$ is not a real momentum since the reciprocal space is an imaginary space which we use for studying diffraction phenomena. Also, $p_1$ is a vector, not an operator
The momentum in the momentum space (just like the position in the position space) has its eigenvectors which will span the momentum space, and its given by $P\left|p\right> = p\left|p\right>$. Also, the momentum vector associated with a wave vector k is given by $- p_{vector} = \hbar \ k$
Also we know that in the position basis, the momentum operator is $-i \cdot \hbar \cdot\frac{\partial} {\partial x}$, and this is derived from differentiating avg of x.
I am trying to find ($i \cdot \frac{\partial} {\partial kx}$) in terms of operator. I am not sure if I can do this. My initial attempt in doing this was to swap the $ kx $ and $X$ since $kx$ is the equivalent of $X$ in position basis, in the momentum space, but seems like its incorrect. Another method, what I did was to equate $p_{vector}$ to the P operator in the momentum space, and arrived at this relation $- kx = -i \cdot \frac{\partial} {\partial x}$. I guess this is also wrong because we cant equate a vector to an operator [Operator acts on a vector]. I am not sure how to proceed with this (unsure if we can even do this).
I need to find this since I am trying to find winding number using quantum computing technique as mentioned in this PRL (https://arxiv.org/abs/2003.06086 - Digital Simulation of Topological Matter using Programmable Quantum Processors). So, for this, (atleast from what I have understood is that) they introduce a displacement operator since its avg over a long time limit is equivalent to the Berry phase and winding number is berry phase mod $2\pi$. The displacement operator in the momentum space is expressed as ($i \cdot \frac{\partial} {\partial x}) \cdot \hat{Z} $ where $\hat{Z} $ is the Pauli Z operator. This acts on the wavefunction to give the required result. I know that I cant do partial derivative wrt kx, unless Pauli Z is operated on the wavefunction, and ($i \cdot \frac{\partial} {\partial kx}$) differentials the transformed wavefunction.
Any assistance is appreciated. I hope this post belongs to the Physics forum.
Edit: I realized that vectors can be operators too, like the divergence operator.
