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In Schrödinger's approach to quantum mechanics, we talk about the probability of finding a particle in a definite location in space. Now if we look at a simple quantum mechanical system, say the behaviour of particle in harmonic oscillator potential, we see that there are nodes, indicating that the probability of finding the particle at those points is zero, indicating that the particle cannot be at that place. This does not make sense to me physically. So I was wandering what can be so special about that point in space that the particle cannot be there?

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  • $\begingroup$ Zero probability events can and do happen. Unity probability events might not occur. The technical nomenclature is "almost never" and "almost certainly", respectively. $\endgroup$ – bright-star Jan 1 '17 at 20:13
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I would not say that the probability of finding a particle in a node is zero. While technically that statement may be correct, it does not make much sense, as the probability of finding a particle in any other point is also zero. It would be more precise to say that the probability density is zero at a node. As for why the probability density in a node vanishes... Well, this is just a consequence of the Schroedinger equation, and the main (if not the only) reason we use this equation is that it correctly describes experimental data. You can use comparisons with the nodes of mechanical standing waves, but I am not sure such a comparison, while useful, explains much.

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  • $\begingroup$ The comparison is useful inasmuch as one understands the solutions to the Schrodinger equation to be generalized standing waves. $\endgroup$ – ZeroTheHero Dec 26 '16 at 1:55
  • $\begingroup$ @ZeroTheHero : I agree that the comparison is useful and I said as much in my answer. Let me just add that solutions of the Schroedinger equations can describe running waves as well, not just standing waves. $\endgroup$ – akhmeteli Dec 26 '16 at 2:42
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This is analogous to the nodes in a plucked guitar string or an organ pipe.

For simplicity let's consider an infinite 1D potential well rather than the quadratic potential of the simple harmonic oscillator. In the well, where the potential is zero, the solutions to the wavefunctions are plane waves, and the plane waves can travel in both directions. When we impose the boundary conditions we find superpositions of the left and right going plane waves give use the well known eigenstates of the particle in the well.

Particle in a box

(Image from Wikipedia)

This is mathematically exactly the same as the standing waves we get from plucking a guitar string or when generating pressure waves in a pipe such as an organ pipe. The nodes are caused by interference of the left and right going waves.

Note that the interference does not remove the particle, it just pushes the particle around. That is it makes the particle more likely to be found in some places and less likely to be found at the nodes. The total probability of finding the particle remains unity.

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    $\begingroup$ OP wants a physical understanding. What's vibrating in the case of a particle in a box? Why, physically, are there points where we never find the particle? $\endgroup$ – DanielSank Dec 25 '16 at 17:13
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    $\begingroup$ The Schrodinger equation is a non-relativistic approximation. The Dirac equation includes spin and relativity and eliminates the points of zero probability density. Nodes still exist for Dirac solutions, however, the probability density is a sum of contributions from both the large and small component solutions and I've never seen a situation where nodes in both the large and small components occur at the same spatial point. $\endgroup$ – Lewis Miller Dec 25 '16 at 20:29

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