Abstract
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.
Highlights
The non-commutativity in quantum mechanics leads to the essential contradistinction between itself and classical mechanics
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
There is hardly any work involving researching Heisenberg-like uncertainty principle via replacing mean value by weak value merely, which is the simplest case in this field
Summary
The non-commutativity in quantum mechanics leads to the essential contradistinction between itself and classical mechanics. Heisenberg uncertainty principle relying on the heritage of weak measurement. There is hardly any work involving researching Heisenberg-like uncertainty principle via replacing mean value by weak value merely, which is the simplest case in this field. We study the Heisenberg-like uncertainty relation in the case of measuring position and momentum in a onedimensional (1D) simple harmonic oscillator with the pre-selected state appointed as one of its eigenstates. Providing the pre- and post-selected states are designed as two distinct coherent states, the variance in the sense of weak values will arrive at the lower bound in usual. This paper is organized as follows: In Section 2 we show our main results about Heisenberg-like uncertainty principle through replacing expectation values by weak values in rudimental.
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