quantum mechanics - When QM particle in a box is in the first excited state, will it be ever found at the middle point, ie. at the nodal point?

Reading the wikipedia article about the particle in the box, there is this image:

Animations from B to F show wave function of a particle in a box starting from ground state up to excited states. The animation C shows wave function behavior in the first excited state and at the middle point both real part (blue) and imaginary part (red) of the wavefunctions are zero all the time. Does it mean that particle will never ever be found at that point? Similarly, for second excited state in the picture D we have two points where this happens, so does that mean that there are two points in space where particle will never ever be found?

I can't really tell from the picture if this continues for higher excited states. Is there some law that says for the n-th excited state there will be n points in space where probability of finding particle is zero?

Answer

The probability of finding a particle at a point is always zero.

Recall that $\rho(x) = \lvert\psi(x)\rvert^2$ is a probability density, not a probability, and so the probability to find the particle somewhere inside the interval $[a,b]$ is given by

$$P([a,b]) = \int_a^b \rho(x)\mathrm{d}x.$$ Since points have zero measure ($\int_a^a \rho(x)\mathrm{d}x = 0$ regardless of $a$), this is always zero for single points. So it is not evident that there is any meaning to saying "the particle will never be found at $x_0$" because quantum mechanics only allows us to talk meaningfully about a region (however small) in which the particle can be found.

This is supported by the fact that the "eigenstates" $\lvert x\rangle$ of the position operator are not actual states since they are non-normalizable ($\langle x \vert x\rangle$ cannot be made finite/well-defined), so there is no actual measurement whose result could be $\lvert x\rangle$, a fully localized particle.

However, you are asking about the nodes of the wavefunction of a particle trapped in a box. Indeed, even though you should not think of them are "points where the particle can never be found", the $n$-th excited state has $n$ of these nodes in its wavefunction.

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garyp suggests an alternative interpretation of "the particle can never be found at $x_0$" in the comments that actually then is correct for the nodes:

For the nodes $n_i$, we have

$$\lim_{a\to 0}\frac{P([n_i-a,n_i+a])}{P([x_0-a,x_0+a])} = 0$$ for any $x_0$ that isn't a node itself. This means, in words, that if we take regions of equal size centered around the points $n_i$ and $x_0$ and shrink them, it becomes more and more likely to find the particle around $x_0$ compared to finding it around $n_i$, until in the limit the ratio becomes zero suggesting it is infinitely more likely to find the particle "at" any other $x_0$ then it is to find it "at" $n_i$. Note that the latter part of this sentence should only be understood heuristically due to the actual probability of finding a particle at a point being zero as discussed at the beginning.