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

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?

• I don't understand what sort of answer except for "They are different because quantum mechanics is really different from classical mechanics" you might be looking for. Additionally, you are comparing apples to oranges here: The quantum mechanical case has complete information about the state of the particle, while the classical probability only arises because you lack information about the particle (namely its position) that you could in principle have. These two situations really are very different, and I don't understand why you would expect the probability distributions to be similar. – ACuriousMind Apr 15 '17 at 11:52
• Yeah I expected an answer like this. A sharper, clearer question that I had was irrespective of the classical probability distribution, why does the particle have a much higher chance of being detected at the middle of the well instead of the edges? Is this something special to an infinite potential well? – Sumant Apr 15 '17 at 13:03
• @Sumant The same occurs with the harmonic oscillator (albeit for different reasons). Note the correspondence principle indicates that quantum and classical distributions must eventually coincide in the limit of large quantum numbers. If you think of the ground state as the least classical state, you can imagine that this is the state for which the distributions will be the most different. – ZeroTheHero Apr 15 '17 at 14:11
• @ACuriousMind there are a lot of cases where quantum and classical aren't that different, so a discussion about why the infinite well is so non-classical could be appropriate. – DanielSank Apr 19 '18 at 22:20
• @DanielSank respectfully I wonder is it necessary to edit Sumant's question so extensively? It seems like some of the edits are not extremely required, and in a few cases do not really make the question clearer. Also there is the side effect that the quotation of the question in my answer is no longer present in the question. I realise that's not important, but it seems like a shame if the edit itself isn't that necessary. Of course there are places where the edit has made it clearly better as well. – user183966 Apr 19 '18 at 22:44