I have two questions regarding (possibly counter intuitive results) Schrodinger equation and its application to two (strictly hypothetical) scenarios.
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Consider the 1D potential $V(x) = – \frac{\alpha}{|x|}$, which is an attractive one. As it is attractive, after a sufficiently large amount of time I'd expect the probability density of finding the particle at $x=0$ (center) to be higher than that anywhere else. I hope it is reasonable to expect such a thing.
But the solution according to Schrodinger equation is (as given in this answer) is $$u_n(x,t) \sim \lvert x\rvert e^{-\lvert x\rvert/na} ~L_{n -1}^1\biggl(\frac{2\lvert x\rvert }{na}\biggr) e^{-E_nt/\hbar}$$ Consider the amplitude at $x=0$ $$\lvert u_n(0,t)\rvert = 0$$ which I think is contradictory to our reasonable expectations based on intuition. I'd appreciate some comments on why it is counter intuitive.
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Consider the particle in box problem and according to Schrodinger's equation, in almost all energy states the probability of finding the particle close to the boundaries is zero.
Now instead of intuition (which is not predicting anything) I'd give a practical example where it is quite contradictory. (Pardon me if this example is not suitable, I am learning and I'd love to know why it isn't intuitively not suitable). Consider a metal conductor, I guess its an example for particle in a box as the potential inside is zero and we know that the charge is usually accumulated at the edges of the conductor.
EDIT
An interesting thing to add. The probability density vanishing at center does not seem to arise in 3-D hydrogen atom. Check the wave function of 1s orbital of Hydrogen atom, it of the form $\psi_{1s}(r) = a e^{-kr}$, where $k$ and $a$ are some constants and $r$ is the radial distance from nucleus.
Best Answer
Question 1:
Your intuition that in the probability density of finding the particle should be the highest at $x=0$ (when it is in the ground state, to be precise) must be correct, but the solution you quoted is faulty.
I read the reference from the answer you linked. There, what the authors really have done is to solve the Schrodinger's equation for $V(x)=- \frac{1}{x}$ and $V(x)=+\frac{1}{x}$ separately (with no restriction on the range of $x$ for either case) to obtain the bound-state solutions $\psi_{+}(x)$ and $\psi_{-}(x)$. Then, they claim that the full solution is such that $\psi(x)=\psi_{+}(x)$ for $x>0$ and $\psi(x)=\psi_{-}(x)$ for $x<0$.
However, what they have solved is a totally different problem than they had originally intended to. Most importantly, the potentials $V(x)=\mp\frac{1}{x}$ have infinite barriers at $x=0^{\mp}$. Therefore, we can expect that a particle bound to the potential $V(x)=- \frac{1}{x}$ ($+\frac{1}{x}$) on the $x>0$ ($x<0$) side can never penetrate into the other side. You can check this fact mathematically by Fourier transforming the momentum-space wave function from Eq. 13 in the reference. You'll see that $\psi_{+}(x)$ completely vanishes for $x\le 0$ and $\psi_{-}(x)$ for $x\ge 0$. This is certainly not what would happen for $V(x)=-\frac{1}{|x|}$.
In the same paper, other people's results are also mentioned. Most notably, some of them apparently obtained eigenfunctions associated with a continuum of negative-energy spectrum, although the authors of this paper consider them unphysical. IMHO, this sick negative-energy continuum probably does exist, and what it tells us is not that we should discard the corresponding solutions while keeping more well-behaved ones, but that Schrodinger's equation for $V(x)=-\frac{1}{|x|}$ is a fundamentally sick problem in the sense that the electron can really fall indefinitely down to the infinitely deep potential well.
Question 2:
Coulomb interaction between electrons in a metal is screened and becomes short-ranged. This is at least a part of the reason why non-interacting electrons in an infinite potential well can be a reasonable model of a metal for certain purposes.
On the other hand, charges accumulating at the boundary of a metal are "excess" charges. Coulomb repulsion between them is not screened. The non-interacting model by no means contains this kind of physics, but may still be used to describe the interior of the metal in which the Coulomb interaction is screened.