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Because there only exist stationary state solutions of the time-independent Schrödinger equation, which correspond to states with single, definite energies, and the energy expectation value is equal to the energy of a given state, does that mean that there is no associated uncertainty with the measurement of the particle?

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If the system is in an eigenstate of the Hamiltonian, such that it solves the time-independent Schrödinger equation, then yes, the energy uncertainty of that state is zero.

It's important to note, though, that for that to hold the system needs to be in that eigenstate for all time, i.e. the $\Delta t$ in the Heisenberg uncertainty principle needs to be infinite. If the system changes at some point (i.e. for some mundane task such as initializing the experiment or performing a measurement to terminate it) then you can no longer state that the system is in that eigenstate solution of the TDSE.

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