Motivation of translation operator

Question

I am reading Shankar's Principles of Quantum Mechanics, in which the motivation for defining the properties of the translation operator is taken from its classical counterpart. Classically, under the regular canonical transformation $$\bar{x}=x+\epsilon~;\bar{p}=p$$ which is an "off-shell" transformation. We get a result about the "on-shell" behaviour of the system, that, if under such a tranformation the Hamiltonian of the system remains invariant, the system has a conserved linear momentum.

When we adapt this to quantum mechanics, we want to define an operator for which $$\langle X\rangle\rightarrow\langle X\rangle+\epsilon;\langle P\rangle\rightarrow\langle P\rangle$$ which is a statement about the expectation values (though I think this is still off-shell) and not the underlying coordinate system that we are using as it was in the classical case (on further analysis the invariance of the Hamiltonian gives us an on-shell result).

My questions are:

1. Do I understand the premise correctly?
2. In the quantum case how do we relate the conditions taken in the form of expectation values to some off-shell condition?

Answer 1

I would expect the translation operator to be more specific than what you describe. Specifically, I would expect the translation operator to change a state localised within some region $\Omega$ to a state localised within a region $\Omega + \epsilon$.

Note that in ordinary quantum mechanics, the probability that a state is within the region omega can be written as $\langle 1_\Omega\rangle$, where $1_\Omega(x)$ is 1 if $x\in \Omega$ and 0 otherwise. In quantum field theory, you will need to combine this function with a density operator, since particle number can vary.

If you manage to single out the operators $1_\Omega$, you can use them to construct a coordinate system. So whether you start with a coordinate system, or with a sensible notion of translation operator, you end op in the same situation in the end.