Position probability distribution of a particle in an infinite square well: classical versus quantum

Question

The wave-function for a single particle in a potential well of width $L$ is given by the relation $$\Psi_n(x)=\sqrt{\frac 2L} \sin(K_n x)$$ where $K_n$ is $(n+1)\pi/L$ and $n$ is a positive integer.

Quantum Mechanical Description of the Probability Distribution

The ground state wave function has $n=0$, so $K_n = \pi / L$ and so $\text{Re}(\Psi_0(x))$ looks like

The probability distribution for this wave function is $|\Psi_0|^2$, which is a $\sin^2$ function and on the same domain the graph is

Classical Mechanical Description of the Probability Distribution

Consider the same problem classically: a particle is in the box with some energy $E$. Classically it has some momentum and it moves around the box, bouncing back and forth off the walls. This particle spends the same amount of time at all points in the box as it is moving with constant velocity. So intuitively, the probability distribution should be constant, i.e. I am just as likely to find it near the wall as the center of the box. Similarly, the probability distribution looks like this:

Why are the two probability distributions so different? In the quantum case, the particle is more likely to be near the center and have a low chance of being near the walls. Classically, the probability should be uniform, so the quantum case is unintuitive. I know that the quantum mechanical description is correct at low energies, but is there an intuitive explanation for why the quantum probability distribution differs so strikingly from the classical one?

Answer 1

As noted in the comments, in one sense the answer is simply "because quantum mechanics and classical mechanics are different theories, some of the predictions will be different".

However, I think it is possible to go farther, particularly with respect to your questions "Why are the two Probability distributions so different?" and "Is classical intuition very bad at lower energies or is it something deeper?".

There are quantum states which are very similar to classical states which are made from combining many energy levels. In the "harmonic oscillator" system, which is quite similar to the "square well" one you have used, these are well studied and called "coherent states". They share many properties with classical mechanics. The wave function looks like a 'blob' that bounces back and forth like a classical particle. If you construct a suitable 'blob' (which will made by adding small amounts of the infinite number of all energy eigenstates) it will bounce around the square well quite similarly to a classical particle. If you integrate/average the probability over a suitable period of time, it will look much more similar to the classical uniform distribution.

So, to answer your question, the two probability distributions are very different because you chose a single energy state. Single energy states are extremely different to classical particle states.

This reasoning also shows why quantum mechanics is better than classical mechanics particularly at low energy levels. At low energy levels, around the bottom few energy states, a particle cannot be in a combination of many states, because we just said it is in the lowest few states only. So therefore we know it is not in one of the states which have more classical properties that are made by combining a great many energy levels, and therefore only quantum mechanics can work well.