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As one of the fundamental traits governing the operation of quantum world, the uncertainty relation, from the perspective of Heisenberg, rules the minimum deviation of two incompatible observations for arbitrary quantum states. Notwithstanding, the original measurements appeared in Heisenberg’s principle are strong such that they may disturb the quantum system itself. Hence an intriguing question is raised: What will happen if the mean values are replaced by weak values in Heisenberg’s uncertainty relation? In this work, we investigate the question in the case of measuring position and momentum in a simple harmonic oscillator via designating one of the eigenkets thereof to the pre-selected state. Astonishingly, the original Heisenberg limit is broken for some post-selected states, designed as a superposition of the pre-selected state and another eigenkets of harmonic oscillator. Moreover, if two distinct coherent states reside in the pre- and post-selected states respectively, the variance reaches the lower bound in common uncertainty principle all the while, which is in accord with the circumstance in Heisenberg’s primitive framework.

Studying Heisenberg-like Uncertainty Relation with Weak Values in One-dimensional Harmonic Oscillator