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Let's consider an infinite square well. In the first exited state there is a node at the middle of the well (i.e. wave function and thus probability of finding the particle is zero there). If I measure the position right now, I may find the particle on the left side of the node, and if I measure the particle after some time, I may find the particle on the right side of the node. But must a particle on the left side to go to right side not pass through the node point? Thus the particle will spend some time at that point, so there should be some probability of finding the particle there, but the wave function is zero there, so what's wrong with the arguement? Does the particle move here to there, without passing through the space in between? But that will just violate continuity equation?

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    $\begingroup$ When you measure the particle the first time, it will stop being in its first excited state, and so afterwards there might not necessarily be a node. $\endgroup$ – Oscar Cunningham Jun 16 '16 at 9:44
  • $\begingroup$ Related: physics.stackexchange.com/q/135520/2451 and links therein. $\endgroup$ – Qmechanic Jan 2 '17 at 14:30
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The wavefunction being zero at a single point has no physical significance because the wavefunction is a probability density (more precisely, its squared modulus is that), not a probability. All that counts are the integrals of the density over regions of non-zero measure, the values at single points are completely irrelevant. Formally, wavefunctions lie in $L^2 (\mathbb{R}^n)$, which are already equivalence classes of square-integrable functions where functions that agree almost everywhere represent the exact same object. Therefore, a wavefunction that is zero that a point is indistinguishable from a function that has any other value there but agrees with it everywhere else.

To address the deeper confusion in your question: A quantum object does not move between two places of detection in the classical sense, a phenomenon sometimes called "quantum tunneling". You simply cannot apply your classical intuition about the motion of point particles to the time evolution of quantum objects.

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