Quantum state where uncertainty in kinetic energy is zero? While reading Shankar's book on Quantum Mechanics, I encountered an interesting problem:

Compute $\Delta T\cdot\Delta X$, where $T = P^2/2m$.

I found several solutions online which arrive at the result $\Delta T\cdot\Delta X \ge 0$.
My question is: does there exist a state $|{\psi}\rangle$ which saturates this inequality, i.e. for which $\Delta T\cdot\Delta X = 0$? We know $\Delta X\ne 0$ (from the uncertainty relation between $X$ and $P$), so then we must surely have that $\Delta T = 0$. But I'm struggling to imagine a physical state with a well-defined kinetic energy! If it is indeed possible, please provide an example of such a state. Thanks!
 A: The minimum value of $\Delta x \Delta T$ is dictated by by the magnitude of the commutator $|[x,T]|=\hbar p/m$.  Only if this vanishes can the inequality be saturated.  This requires a momentum eigenstate with $p=0$, which means a constant wave function $\psi=c$.
Whether such a wave function can exist is a trickier question.  Obviously, the domain of the wave function must be finite, or else $\psi$ will not be normalizable.  If the particle is confined to a finite region by a potential, then $p=0$ will not be an energy eigenstate.  This leaves us with the case of a finite region with periodic boundary conditions.
For a real particle, if $x$ is the Cartesian coordinate, this is not a realistic arrangement.  (Periodic boundary conditions are often used in statistical mechanics, but there they are idealizations that are useful when the thermodynamic limit is taken.)  However, if $x$ represents something like the position of a bead on a hoop, then the zero-momentum state is the real physical ground state.  More generally, such a state will exist when the coordinate variable represents an angular position.  For instance, in three dimensions, an S state wave function in a central potential has zero angular momentum and zero angular momentum uncertainty, although there is still position-energy uncertainty, because of the radial dependence of the kinetic energy.
