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I have been looking at the quantum harmonic oscillator, or at least the lowest energy level anyway and came across an interesting feature. The second position derivative of the wave-function is 0 at the edge of the classically allowed region.

$$ \phi\propto e^{-x^2/2a^2} \implies \frac{d^2}{dx^2}\phi(x=\pm a)=0 $$

where $a=\sqrt{\frac{\hbar}{m\omega}}$

The total energy is $E=\frac{\hbar\omega}{2}$ and the potential energy is $V=1/2 m\omega^2x^2$. These are equal when $x=\pm a$.

Why do the vanishing of the second derivative and the boundary of the classically allowed region occur at the same point?

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For an energy eigenfunction of energy $E$: $$0 = (\hat{H} - E) \,\psi = \frac{1}{2m} \hat{P}^2 \,\psi + (\hat{V}-E) \,\psi$$ or, in position representation: $$\frac{1}{2m} \frac{d^2}{dx^2} \psi(x) = (V(x)-E) \,\psi(x)$$ so the second position derivative of $\psi$ vanishes whenever $V(x) = E$, ie $x$ is at the edge of the classically allowed region.

As mentioned by @By Symmetry, we can also read from this equation that the wavefunction tends to have an oscillatory behavior* inside the physically allowed region, and be exponentially suppressed outside.

* In the case of the groundstate of the harmonic oscillator, the oscillatory behavior is not obvious because there is only $1/2$ an oscillation inside the allowed region...

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The ground state wavefunction of the harmonic oscillator is a Gaussian. The inflection points of a Gaussian (where the second derivative is $0$) occur at plus and minus one standard deviation from the mid-point. So this is, slightly indirectly, telling you that the average spread of the position of the particle in the ground is given by the size of the classically allowed region.

It is a general feature that in classically forbidden regions the kinetic energy term is the Hamiltonian must be negative, which tends to result in the wavefunction decaying exponentially in these regions.

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At a turning point, or inflection, where the wave equation changes from an upward moving point through an infinitely short (presumably) change to a downward moving path, the inflection seems to impose the condition that an almost bewildering variety of different paths may begin. The wave, at the inflection, does not know whether it is continuing a sine wave, hyperbola, straight line, circle, ellipse, or any other smooth continuous curve.

I saw a group of the possibilities within a photon wave. It was either an energy or a momentum wave. I used ideas like the Planck Time and Planck Length to get a sense of the wave's image. Anyway, the instant of ambiguity at the inflection seems to be where uncertainty may arise as to frequency, momentum or whatever other variable the wave is following, will do.

At that point, it appears the wave loses a very minute part of it's true nature into the probability that it is one of SEVERAL possibilities a,d the ambiguity results in that minute loss which may be a part of, say, a quantum theory of the observed red shift in light from extremely distant galaxies.

Again, in that instant he possibilities apart from the true path (which eventually of course dominates) includes ellipse, a circular paths in the wrong direction, a hyperbola, gaussian, etc. It helps to become familiar with the Planck action quantum, the Planck mass, Planck time, Planck length and other small constants, because most of the waves we visualize are in light or sometimes radio-tv waves.

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