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please note that I am a high school student trying to understand the quantum model of the atom; I have only the most basic understanding of quantum mechanics.

I am trying to comprehend the wave nature of electrons in atoms but I am unsure on a number of things.

  1. what is meant by a probability cloud of electron(s)? does this mean that the electron is rapidly moving inside the probability cloud/ orbital, present in some regions for a longer time than in others, or is it sort-of smeared throughout the orbital until it is observed?

  2. if an electron exists as a stationary wave around an atom, then what is "waving"? mechanical stationary waves, for example, consist of displacement oscillating with amplitudes that vary with position. What physical quantity is actually oscillating here? Surely, the electrons themselves aren't moving up and down, are they?

Edit:

the following excerpt is taken from https://en.wikipedia.org/wiki/Atomic_orbital

Wave-like properties:

  1. The electrons do not orbit the nucleus in the manner of a planet orbiting the sun, but instead exist as standing waves. Thus the lowest possible energy an electron can take is similar to the fundamental frequency of a wave on a string. Higher energy states are similar to harmonics of that fundamental frequency.

  2. The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron. The charge on the electron acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's wave function.

IF theese standing waves are probability waves, how can they be related to the discrete energy levels? morever, how can the charge be smeared out if the electrons themselves are not?

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Let's first discuss your first question because it forms the meat of the confusion.

The probability cloud of an electron is simply the function $\vert \psi(\mathbf{r})\vert^2$ which gives you the probability density at $\mathbf{r}$ corresponding to the probability of finding the electron in a given region of space upon a measurement of the position of the electron.

This probability, while very similar to the probability that shows up in classical statistical mechanics, is different than the classical statistical probability in the following crucial sense: in classical statistical mechanics, when one would talk of the probability of finding the electron somewhere, they don't mean that the electron is actually not at any particular place before the measurement. In quantum mechanics, we do mean that. So, your guess that the electron spends more time in one place than the other would make perfect sense if we were talking about probabilities of a particle in classical mechanics. However, in quantum mechanics, that is not the case. In fact, the electron simply doesn't have any position unless you measure it. So, it is also not true that the electron is somehow spread out or has multiple positions at once.

The crucial insight of quantum mechanics is that one cannot associate all physical quantities in a well-defined manner with a particle at once. Certain physical quantities are incompatible with each other, this means that if a particle has a well-defined value of the first quantity, it doesn't have a well-defined value of the other quantity. What we do have is the probabilities for the different values of that other quantity that we will get when we measure that quantity for the given particle. The most famous pair of incompatible quantities is the pair of momentum and position. A quantum particle cannot have a well-defined position and a well-defined momentum at the same time. However, for a free quantum particle, the energy and momentum are compatible so a free particle can have both a well-defined momentum and well-defined energy at the same time.

So, in conclusion, the probability cloud is what it is, the probability cloud. And that probability is different from the classical probabilities in the sense described above.

Your second question is much simpler to answer. I will reproduce what I wrote in one of the comments:

"Wavefunction" is simply a name, don't read into it too much. Take the actual definition of the wavefunction more seriously than its name. If the electron is in an energy eigenstate, nothing physical about the electron would be changing at all with time (the wavefunction would pick up an overall phase factor (𝑒−𝑖𝐸𝑡 where 𝐸 is the energy and 𝑡 is time) but that is physically irrelevant and no measurable quantity depends on this overall phase factor.).

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  • $\begingroup$ could you please check out the edit? How can the electric charge be spread out if the electron itself is not? $\endgroup$ Oct 21 '20 at 17:11
  • $\begingroup$ @OVERWOOTCH It doesn't say that is actually spread out, it is trying to qualitatively give an idea that it acts as if it is spread out. If one were to say that it is actually spread out, they would be simply wrong. One doesn't find a part of the electron anywhere. One always finds the full electron. $\endgroup$
    – Dvij D.C.
    Oct 21 '20 at 17:24
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what is meant by a probability cloud of electron(s)? does this mean that the electron is rapidly moving inside the probability cloud/ orbital, present in some regions for a longer time than in others, or is it sort-of smeared throughout the orbital until it is observed?

Neither. There isn't anything underlying the probabilities in Quantum Mechanics. This is in contrast to say, usual probability where we use probability to deal with things we do not know about the system. It's not like we are saying "the electron could be here, we just don't know." Rather, we actually cannot say anything about the position of the electron until we measure it. Therefore, this throws out the idea that it stays in some regions longer than others since it isn't in any region before measurement.

The idea of it being "smeared out" also doesn't make sense. Electrons are point particles; you can't smear them out.

if an electron exists as a stationary wave around an atom, then what is "waving"? mechanical stationary waves, for example, consist of displacement oscillating with amplitudes that vary with position. What physical quantity is actually oscillating here? Surely, the electrons themselves aren't moving up and down, are they?

This is a mix up with the idea of the wave function of the electron. However, the electron itself isn't the wave function; the wave function just describes the state of the electron. The electron isn't existing as a wave itself; electrons are particles.

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  • $\begingroup$ is it the probability that is oscillating? if yes, what quantity is it oscillating in? (time. space....) $\endgroup$ Oct 21 '20 at 16:17
  • $\begingroup$ "However, the electron itself isn't the wave function; the wave function just describes the state of the electron." This is not a trivial statement, one would be very well justified in saying that the electron is simply its state vector(ray) in the relevant Hilbert space. $\endgroup$
    – Dvij D.C.
    Oct 21 '20 at 16:21
  • $\begingroup$ @DvijD.C. In classical mechanics would one say a particle is its vector in configuration space? $\endgroup$ Oct 21 '20 at 16:22
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    $\begingroup$ @DvijD.C. I guess I don't like equating physical objects to the mathematical objects that describe them. I feel like saying the electron is the wave function is nonsensical. The wavefunction mathematically can be defined over all space. Does this mean the electron exists over all space? If so, then when we measure the electron to be somewhere what is the mechanism that collects all of the parts of the electron across all space and brings it to the measurement point? $\endgroup$ Oct 21 '20 at 16:27
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    $\begingroup$ Fair enough, I tend to oscillate between the two positions but all I am saying is that neither position seems trivially correct/incorrect to me. In any case, that is irrelevant because even if the wavefunction simply describes the electron and is not the electron itself then the question still remains as to what physical aspect of the electron is changing with time if this mathematical representation of the physical object is changing with time. $\endgroup$
    – Dvij D.C.
    Oct 21 '20 at 16:29
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We do not know what the probability cloud actually is, all we know is that if you look for the electron, the equation for the cloud tells you how likely you are to find it in different places. That's how it gets its name. There has always been a lot of argument about whether it might be an actual particle or something sort of smeared out or something much more weird, so the convention among physicists is to simply ignore the question. If you are taught "probability cloud", best to settle for that until your teacher tells you otherwise!

A "stationary wave" is also often called a "standing wave". It means that the wave appears not to be moving, even though it is shuttling back and forth over exactly the same region all the time. The vibration of a guitar string is a more everyday example. But again, the thing that is doing the waving is that mysterious whatever-it-isn't that turns up somewhere-or-other.

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  • $\begingroup$ I am aware of mechanical stationary waves where the the oscillating quantity is displacement from the equilibrium position against time. What is the oscillating quantity here and what is it oscillating with? $\endgroup$ Oct 21 '20 at 16:28
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    $\begingroup$ -1: This is a misleading answer IMO. One can argue as to whether the standard version of quantum mechanics is the best or not, but the standard version of quantum mechanics (the one that is taught to every physicist) has clear answers to everything the OP is asking. The electron is neither smeared (because that would mean there is a little bit of electron everywhere) nor is it that the electron spends more time in certain places than others. The answer is that the electron simply doesn't have a position unless a measurement of the position operator has just been made on the electron. $\endgroup$
    – Dvij D.C.
    Oct 21 '20 at 16:36
  • $\begingroup$ @DvijD.C. Copenhagen does not deny that the electron is smeared out or has a position, it regards those suggestions as meaningless nonsense - just as it regards yours. That is to say, any discussion - including denial - of the electron's position before measurement simply has no meaning. Yes a textbook answer would ignore the nonsense and jump straight to measurement, but the OP asked a bit about the nonsense so it makes sense to provide a brief summary. I'll think about making that a bit clearer. $\endgroup$ Oct 21 '20 at 16:56
  • $\begingroup$ @OVERWOOTCH The best thing is just to think of it as an electron and leave it at that. University students are taught that it is a perturbation of the zero-point electron field, but that will not help you get through high school! As you can see, some folks already think I have gone too deep into some of the things you asked about. $\endgroup$ Oct 21 '20 at 17:05
  • $\begingroup$ @GuyInchbald That is simply not true. Every textbook clearly says that in quantum mechanics, a particle is said to have a well-defined property X if it is in an eigenstate of the operator X. Also, every good textbook talks of quantities before measurement all the time. For example, any book that discusses the eigenstates of the Hamiltonian tells you that if a particle is in the nth energy eigenstate, its energy is $E_n$. You can measure it and nobody would be hesitant to say that it was $E_n$ even before. $\endgroup$
    – Dvij D.C.
    Oct 21 '20 at 17:08

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