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Consider electron in an infinite potential well, studied in quantum mechanics. Position probability density of the electron is

$$ P_n(x)=\left(\frac{2}{L}\right)\sin^2\left(\frac{n\pi x}{L}\right)$$

where $0\leq x\leq L $ and $L$ is length of the box.

So for $n>1$, probability density & hence probability of finding location of the electron at certain $x$ is $0$. The electron moves from left to right & right to left between the walls of the well. So Mathematics says that electron can not be found at certain $x$ node locations within the box; which is very strange. But is there any experimental evidence for this? While crossing these special $x$ node locations, as if electron disappears from the box. This is very absurd.

My question is: Is this a just mathematical result (without any reality) or a physical reality/actuality?

Sebastian Riese
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atom
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1 Answers1

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You are imagining the particle in the well as a classical system i.e. a point particle moving to and fro in the well. However this is not a good description of the system. A quantum particle does not have a position. By this I mean that it is meaningless to ask what the position of the particle is because position, in the sense we normally use the term, is an emergent property of a macroscopic system.

Instead the particle has a probability distribution that tells us the probability of finding the particle in some infinitesimal volume element. This probability distribution falls to zero at some places, but that doesn't mean the particle disappears as it passes through those places.

As for experimental evidence, the obvious example of a particle in a (finite) potential well is an electron in a hydrogen atom. The atomic orbitals have nodal planes where the probability distribution falls to zero just like the hypothetical particle in an infinite potential well. These nodal planes were directly imaged in 2013 by Stodolna et al. A description of the paper can be found here.

John Rennie
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  • By experimental evidence I mean that is there any human who has tried to find experimental probability to find location of electron in Hydroegen atom?Is there any published literature on such experiments? – atom Jul 20 '15 at 11:05
  • Isn't your statements contradictory - (1)"A quantum particle does not have a position" and (2) "particle has a probability distribution that tells us the probability of finding the particle in some infinitesimal volume element" ?......probability distribution which is a continuous curve has no meaning unless you consider electron to be of finite size i.e. of the order $10^{-10}m$. Am I wrong ? – atom Jul 20 '15 at 11:15
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    @atom: I don't understand your argument that the particle has to have a finite size to have a probability distribution. In principle the electron in a hydrogen atom has a non-zero probability of being a metre from the atom. Does that mean the electron is a metre in size? Obviously not. – John Rennie Jul 20 '15 at 11:31
  • @atom: for experimental evidence see this. I've updated my answer with the details. – John Rennie Jul 20 '15 at 11:34
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    @atom, John's statements aren't contradictory; a quantum particle does not, in general, have a definite position until a position measurement is made. The energy eigenstates of the infinite well are not states of definite position. Further, when a position measurement is made, the particle is no longer in an energy eigenstate, i.e., the measurement process results in the state changing from one of definite energy to one of definite position. – Alfred Centauri Jul 20 '15 at 12:10
  • @AlfredCentauri, Consider 2 statements (s1) The s,pd,f orbitals of the atom gives electron's position probability density inside the atom. (s2)There are nodal points in these orbitals at which electron never resides. s1 does not imply electron is smeared/spread within orbital's volume. s2 compells us to think (logically) this way -> there are 3 distinct things: probability density, nodal points, and electron. ....continued in next immediate comment.... – atom Jul 21 '15 at 04:27
  • @AlfredCentauri, So now question comes what is electron then ? If you say it is not a particle then it must be some smeared-out-thing. But then probabilty density will have no meaning; since each point of that density or orbital corresponds to some specific (x,y,z) or ($r,\theta,\phi$). Our smeared-out-thing has many density values then at one instant of time! – atom Jul 21 '15 at 04:27