Derivation of <PERSON>'s theorem of the first-form
I am currently working through <PERSON>'s, Quantum States of Atoms Molecules and Solids, and am having some trouble understanding his derivation of <PERSON>'s theorem of the first form.
We begin with the notion that the Hamiltonian is invariant under transformations for a periodic potential, $V(r)=V(r+T)$, where $T$ is the lattice translation vector, $T=n_1\textbf{a}+n_2\textbf{b}+n_3\textbf{c}$, such that $H(r)=H(r+T)$.
My first question arises here: Why can we assume that the kinetic energy term is invariant under translation?
Regardless, assuming it is invariant under translation for the purpose of this question, we define a translation operator, $\hat{T}$, such that, $\hat{T}f(r)=f(r+T)$, and that $\hat{T}$ can be represented by a function $T(n_1,n_2,n_3)$. Consequently, there are an infinite number translation operators, indexed by $n_1,\,n_2$, and $n_3$. We may hence define a second translation operator, $\hat{T}'$, indexed by $n_1',\,n_2'$, and $n_3'$, such that $\hat{T}\hat{T}'=\hat{T}'\hat{T}$, and therefore, that $[\hat{T},\hat{T'}]=0$.
The book goes on to define $\hat{T}''=\hat{T}\hat{T}'$, corresponding to the translation vector, $T''$, indexed by $n_1+n_1',\,n_2+n_2'$, and $n_3+n_3'$.
Due to (the presumed) translational invariance of the Hamiltonian, we also know that $[H,\hat{T}]=0$, for any $\hat{T}$ (including primed $\hat{T}$s), such that $\hat{T}\psi=c\psi$, $\hat{T}'\psi=c'\psi$, and $\hat{T}''\psi=c''\psi$, where $\psi=\psi_E$ are the energy eigenfunctions. 474
Imposing normalization conditions, we can say that $|c|^2=|c'|^2=|c''|^2=1$. The book at this point goes on to write these constants in the form, $e^{i\beta},\,e^{i\beta'}$, and $e^{i\beta''}$. Because information cannot be lost $\beta,\,\beta'$, and $\beta''$, must be indexed by $n_1,\,n_2,\,n_3$, $n_1',\,n_2',\,n_3'$, and $n_1+n_1',\,n_2+n_2',\,n_3+n_3'$, respectively.
This gives us, $\beta''(n_1+n_1',\,n_2+n_2',\,n_3+n_3')=\beta(n_1,\,n_2,\,n_3)+\beta'(n_1',\,n_2',\,n_3')$. At this point, the book states the following:
The point to be emphasized is that $\beta''$ is the sum of $\beta$ and $\beta'$, and depends on the sums of the corresponding indices, such that for the above equation to hold, $\beta$ must be proportional to $T$, where $T$ is the lattice translation vector, such that $\beta\propto T\to\beta=k\cdot T$, where $k$ is not dependent on the indices.
I am simply not following the jump in logic here.
Why must $\beta=k\cdot T$ for $\beta''(n_1+n_1',\,n_2+n_2',\,n_3+n_3')=\beta(n_1,\,n_2,\,n_3)+\beta'(n_1',\,n_2',\,n_3')$ to hold? 610