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I used to think that the electron is a particle orbiting the nucleus, but now I know that the electron can be also thought of as a standing wave.

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That's kind of like saying that a curve is both straight and circular.

I've also learnt that if you mark the position of the $p$ electron over time, you get

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So $90\%$ of the time, the electron is inside the dumbbell. I can't see how that relates to the idea of an electron being a standing wave!

Does anyone really understand this?

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Electrons are elementary particles and thus are quantum mechanical entities. When studying electrons one should stop thinking classically, either classical waves or classical particles, and acquire a quantum mechanical intuition based on the mathematics of QM.

QM tells us that to every observable, like momentum, position and the other quantities we have defined macroscopically, there corresponds an opertor. This is a differential, most of the time, operatorx acting on the wave function of the system.

QM tells us that the wave function of the system squared gives the probability of finding the particle under study at an (x,y,z) at time t, and if the wave function is operated upon by one of the QM operators it gives the probability distribution for the observable under consideration.

These two postulates are critical in interpreting the data with the solutions of the wave type differential equations of QM. The waves are not energy or matter waves as with water or sound . They are probability waves, i.e. the distributions are probability distributions. The p orbital you display is a probability distribution for finding the electron at that space point. Probability distributions mean the same classically and quantum mechanically. The probability distribution for tossing a dice would be a flat distribution. For QM set ups it is more complicated , but the idea is the same: "what is the probability of the dice coming up six" and "what is the probaility of the electron being found in (x,y,z)" are read off from the corresponding distributions.

The standing wave interpretation was proposed before quantum mechanics was established as the correct mathematical theory for elementary particles,when the atom was thought of classically as a planetary system, the Bohr atom. Although it fitted the data, it was not a theory, just a model. The theory of QM explains both the structure of the atoms and the spectra and goes much further in describing and predicting behaviors in the microcosm . It is confusing to mix the outdated Bohr atom interpretation with the QM interpretation, this is because it is connected with classical energy and matter waves, and is a wrong frame. It is the probability distributions that display wave like properties in space. Not the particle's mass or energy.

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In answer to your last question, no-one understands this and no-one should pretend that they do. Human beings imagine (or model) electrons as points, waves, strings, vibrations etc. depending on how we are measuring them.

To illustrate this even further, it sometimes appears that a fundamental particle can be everywhere in the Universe, or take every path, at the same time!

All that one can say is that one understands how to apply the current theory, however we can't ever say that a particular theory is 'correct', even if it has the power to predict. We can only say that it fits the data. For instance, the Copenhagen Interpretation of quantum phenomena is extremely successful but it is just that, an interpretation.

So, we can think of the electron any way we like. In some ways, Rutherford's 'plum pudding', Bohr's orbits and shells etc. can still be just as useful as a model of something that we can't see.

I hate to labour the point but all mathematics is abstract modelling. It works, but we don't know why!

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