If you look at the wave function of a particle in infinite square problem for some specific energy level, say for n =1, then the probability of particle to be found in middle of the well is higher than at any other point. Similarly for higher energy levels, there are points called nodes where the particle can't be found. What is the physical interpretation for this? Why are some points more probable than others?

  • $\begingroup$ I'm assuming the current answers are missing the point of your question. You're not asking about why QM theory tells us there are nodes. You are instead asking if there is something "deeper" that tells us why these nodes should arise. Is this correct? $\endgroup$ – BioPhysicist Dec 14 '19 at 15:20
  • $\begingroup$ Yes it's true, because I know that by putting the suitable boundaries condition and using the Schrodinger equation you get the solution that all this to you, Is there anything deep or that's all it is? $\endgroup$ – Young Kindaichi Dec 14 '19 at 17:07

I am going to focus on the last sentence of your question, which sums it up: "why are some points more probable than others?"

The answer is that the waves have to satisfy Schrodinger's equation, and that equation includes that higher kinetic energy goes with higher $d^2 \psi/dx^2$. Meanwhile the boundaries of the box exert their influence, which is that the wavefunction has to go to zero there. So the overall shape of the wavefunction is a combination of these two properties. The maths here is essentially the same as that which describes standing waves on a classical string such as the string of a violin or guitar. In the case of the violin, each part of the string is the same as other parts, but when there is a vibration all at a single frequency, then there is a fixed wavelength, and there is only one way to fit these waves into the region between the two ends. Similarly, for the quantum particle/wave in a box, for a given energy one has focussed on a solution where all parts of the wavefunction oscillate at the same frequency, and this implies a fixed wavelength, and the ends determine how the waves fit into the box. It is the combination of these features which produces the nodes and hence the fact that if one were to detect the location of the entity (particle) then one is more likely to find it in some places than others.

You should note that the presence of nodes at a fixed places is a special property of the states of well-defined energy. But the entity doesn't have to have a completely precisely defined energy. If it is in a superposition of energy eigenstates then its wavefunction will not necessarily have these nodes, except at the walls.

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  • $\begingroup$ Is it worth making special note that in the correspondence principle limit the structure is finer than any reasonable experimental resolution and so in in unobserved? $\endgroup$ – dmckee --- ex-moderator kitten Dec 16 '19 at 3:46

I will not go any deep into the nature of reality but instead I will try to convince you why it has to look like from a logical point of view:

-The potential is infinite at the edges of the well. Therefore, we have to agree that the probability to find the particle there is zero. These are the nodes of the wavefunction squared.

-We also have to agree that the particle exists somewhere in the well, so the wavefunction cannot just be zero everywhere inside.

-Probabilities cannot be negative so, if the wavefunction squared is to be non-zero within the well, it has to be positive. It also has to be smooth since it wouldn't make sense that the probability changes abruptly from one point to another: there is nothing special about any point that would lead to this.

-The whole system is symmetric with respect to a vertical line in the centre of the well. There is nothing special about either side.

-If we put all of that together, we end up with something that looks like:

enter image description here

If you want to then understand why sometimes some nodes appear within the well, then you need to learn some quantum mechanics. Or simply read about standing waves and then realise the wavefunction of the electron within the well is just a standing wave with the contraint of having nodes on the limits of the well.

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