# Basis for the two particle Hilbert space

In Shankar's QM book Chapter 10 pg. 248, it was said that there are other bases possible besides $$|x_1x_2\rangle$$ for the two particle hilbert space. For example, the momentum basis, consisting of simultaneous eigenkets $$|p_1p_2\rangle$$ of $$P_1$$ and $$P_2$$.

More generally, we can use the simultaneous eigenkets $$|w_1w_2\rangle$$ of two commuting operators $$\Omega_1(X_1,P_1)$$ and $$\Omega(X_2, P_2)$$ to define the $$\Omega$$ basis, since any function of $$X_1$$ and $$P_1$$ commutes with any function of $$X_2$$ and $$P_2$$.

Did he miss out saying that the operators $$\Omega_1(X_1,P_1)$$ and $$\Omega_2(X_2,P_2)$$ need to be hermitian? Since two commuting Hermitian operators share a same eigenbasis.

Also, can the $$\Omega_1$$ and $$\Omega_2$$ operators be for example $$X_1$$ and $$P_2$$? Does it need to be the 'same' operator but acting on different particles, e.g. $$X_1$$ and $$X_2$$, $$P_1$$ and $$P_2$$?

A two-particle Hilbert space is just a tensor product $$H\otimes H$$ for the one-particle Hilbert space $$H$$.
Essentially by definition of the tensor product, given any two bases $$\{b_i\}_{i\in I}$$ and $$\{c_i\}_{i\in I}$$ of $$H$$ (with $$I$$ an index set whose cardinality is the dimensionality of $$H$$), the simple tensors $$\{b_i\otimes c_j\}_{i,j\in I}$$ form a basis of $$H\otimes H$$. So in particular you get a basis of $$H\otimes H$$ by picking two arbitrary self-adjoint operators on $$H$$ and choosing their eigenbases for $$b_i$$ and $$c_i$$.
There is nothing special about the specific situation of $$H = L^2(\mathbb{R})$$ in quantum mechanics. You can choose any two bases you want, in particular the eigenbases of two different self-adjoint operators.
The remaining problem in your question is what $$\Omega_i(X,P)$$ is even supposed to mean. You can define a function $$f(O)$$ of a single self-adjoint operator $$O$$ via functional calculus without problems, but what is a function of two non-commuting observables like $$\Omega_i(X,P)$$ supposed to mean? Without further elaboration this is simply bad notation.
One possible interpretation is that the $$\Omega_i$$ are supposed to be polynomials, and that the ordering ambiguities in applying a polynomial to non-commuting operators are resolved always in the fashion that yields a self-adjoint operator (e.g. the function $$f(x,p) = xp$$ is turned into the self-adjoint operator $$\frac{xp+px}{2}$$). In that case it would be superfluous to "restrict" such $$\Omega_i(X,P)$$ to self-adjoint possibilities, since they would always be self-adjoint.