  We point out that the Gaussian wave-packet formalism can serve as a concrete
realization of the joint measurement of position and momentum, which is an
essential element in understanding Heisenberg's original philosophy of the
uncertainty principle, in line with the universal framework of error,
disturbance, and their uncertainty relations developed by Lee and Tsutsui. We
show that our joint measurement in the Gaussian phase space, being a Positive
Operator-Valued Measure (POVM) measurement, smoothly interpolates between the
projective measurements of position and momentum. We, for the first time, have
obtained the Lee-Tsutsui (LT) error and the refined Lee error for the
position-momentum measurement. We find that the LT uncertainty relation becomes
trivial, $0=0$, in the limiting case of projective measurement of either
position or momentum. Remarkably, in contrast to the LT relation, the refined
Lee uncertainty relation, which assesses errors for local representability,
provides a constant lower bound unaffected by these limits and is invariably
saturated, for a pure Gaussian initial state. The obtained lower bound is in
agreement with Heisenberg's value.
