# Are atomic orbitals electron standing waves?

Do electrons really exist as standing waves in an atom? If so, are these standing waves what are known as atomic orbitals?

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ShayanJ
Gold Member
Consider a mischievous boy who is an elementary school student. Every morning this boy walks to school alone and after school, walks to home alone. But he doesn't just take the path to home. He always wanders a bit.
Imagine you go to his mother and ask whether she knows where her son is. She looks at the clock and says the school is finished ten minutes ago and so he should be on the way. But where? She can't tell you where her son is, only where he is probably. So she brings a map and marks different regions where she thinks the boy may be in. Those regions are called orbitals.

bhobba
Mentor
Do electrons really exist as standing waves in an atom? If so, are these standing waves what are known as atomic orbitals?
No.

They are solutions to Schroedinger's equation.

The why of Schroedingers equation is actually quite deep, involving some tricky advanced math - if you are interested you will find it in Chapter 3 of Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20&tag=pfamazon01-20

Believe it or not its the principle of relativity - the POR.

Thanks
Bill

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When one says that the electrons in an atom are in stationary states (or stationary eigenstates), he means that their state is stationary, meaning that it does not change in time. If they are in a certain state (e.g. they are located around some point in space) you can be sure that if you look after a certain time, they will still be where (or how) they where before.

One then talks of atomic orbitals because even those states (that by definition characterize completely all the "characteristics" an electron can have, like position, momentum, spin and so on) do not tell you exactly where the electron is, or what is its velocity. Instead they give you a "probabilistic estimate" of where the electron can be, or of what its velocity can be (like Shyan very nicely described).

Here is a nice site that allows you to visualize the atomic orbitals of hydrogen.

jtbell
Mentor
Do electrons really exist as standing waves in an atom? If so, are these standing waves what are known as atomic orbitals?
Mathematically, the hydrogen atom “orbitals” (energy-eigenstate solutions of the Schrödinger equation) are very similar to standing waves of e.g. sound in a spherical cavity.

For standing waves of sound in a spherical cavity, the microscopic displacement of an air molecule at location ##(r,\theta,\phi)## as it oscillates has the form
$$f(r,\theta,\phi,t) = R_n(r)Y_{lm}(\theta,\phi) \cos (\omega t)$$
where ##\omega = 2 \pi f## and n, l, m are integers which label different forms of the R and Y functions.

The hydrogen orbitals have the form
$$\Psi(r,\theta,\phi,t) = R_n(r)Y_{lm}(\theta,\phi) e^{i \omega t} = R_n(r)Y_{lm}(\theta,\phi) \left[ \cos (\omega t) + i \sin (\omega t) \right]$$
where ##\omega = 2 \pi f = 2 \pi E / h = E/\hbar## and again n, l, m are again labels which we usually call “quantum numbers.”

The R functions are different for the two situations, because the boundary conditions are different. The sound waves do not penetrate beyond the wall of the spherical cavity, but there is no "wall" that sets a fixed boundary to the hydrogen ψ function.

However, the Y functions are exactly the same: the well-known and well-studied "spherical harmonic" functions.

And the sound waves oscillate in time according to a real cosine function, whereas the hydrogen ψ oscillates according to a complex ##e^{i\omega t}##.

As you probably know, the complex "square" ##|\Psi|^2 = \Psi^*\Psi## gives you the relative probability of finding the electron in a particular location ##(r,\theta,\phi)## at time t, if you actually make such a measurement.

If you don't make the measurement, there is no generally accepted answer to the question, "what is the electron 'really doing' in the meantime?" The mathematics of QM doesn't address this question. This is the province of interpretations of QM. People argue about them endlessly because there are few ways to address them with experimental data.

Questions like "Do electrons really exist as standing waves in an atom?" lead down a rabbit-hole into a maze of twisty little passages, where people ask questions like "what do you mean by 'really' and 'exist'?" :tongue:

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vanhees71
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