# Quantum Mechanics - How do we know that the observed locations of electrons are random? [duplicate]

How do we know that the observed location of a electron (or any quantum object) is purely random (there is no way to predict it) within the probability-function instead of normal randomness (we don't know how to predict it)?

As an analogy if someone would only be able to measure the amount of people (amount of electrons) and their individual IQ (location) he would see a perfect distribution (obviously only if he has enough samples) within a specific range (probability) and each individual measurement seems purely random (QM-randomness). He would then assume that their IQ is purely random as he has just not enough knowledge to prove otherwise (he hasn't spoken to any of them, hasn't analysed any in respect to education nor has he analysed their DNA).

Could the same be happening when we observe electrons or have we proven that their location is purely random?

• Possible duplicate of How do we know that certain quantum effects are random? – Stéphane Rollandin Jun 4 '19 at 12:23
• Is this an empirical question? Or is this a question about quantum theory? If it's an empirical question, then what does "random" mean empirically? The usual meaning is "we don't know how to predict it, except for the distribution," but then the question is asking "How do we know we don't know how to predict it?" – Chiral Anomaly Jun 4 '19 at 12:41
• @ChiralAnomaly Thanks for your response as this is very fundamental to my question. With random I mean "There is no way to predict it" and not "We don't know how to predict it". If it's random the way you define it, my question would be answered. My question arised as I've always been told that in quantum mechanics there is this "true randomness" with nothing causing it. I want to know if this truly is a "true randomness" and if we have proven so. – Leander Studer Jun 4 '19 at 12:47
• @Leander BTW, those things like education & DNA correspond to local hidden properties of the electron which determine its location but which we are unable to measure for some reason. But the Bell inequalities tell us that the electron doesn't actually have any such local hidden properties, because there is no way to assign values to those properties which is mathematically consistent with the observed results in Bell-type experiments. – PM 2Ring Jun 5 '19 at 9:31
• @PM2Ring Thanks a lot for you comment, you pointed me into the right direction. So we are only certain of the fact that there are no local hidden variables, but not non-local hidden variables? If you write your explanation as an answer I'll accept it. Maybe mention that it's not possible for local hidden variables to exist while the possibility for non-local hidden variables is still there. – Leander Studer Jun 5 '19 at 14:52

The location of an electron is not random by any means, rather it is exactly determined by the coefficients of its expansion onto a basis. Namely let the state of the electron be described by $$|\psi\rangle = \sum_n c_n |a_n\rangle$$ where $$|a_n\rangle$$ is a complete basis of the Hilbert space and eigenstates of an observable $$\hat{A}$$. This means that if we perform infinite measurements of the physical quantity associated to $$\hat{A}$$, we found the value $$a_n$$ exactly $$|c_n|^2$$ times each. Replace $$\hat{A}$$ with the position operator and the coefficients will represent the probability of finding the particle in that position if we make infinite measurements.