Do I get shells, subshells, and orbitals right?

Question

I would like to make sure my (very superficial) understanding of electron shells, subshells, and orbitals is correct. I did look them all up on the Internet, but none of the sources said exactly what I think is going on, so I might be wrong.

So, as far as I understand, shell, subshells, and orbitals are not, strictly speaking, actual, physical locations around an atom's nucleus. Shells and subshells are more like abstractions: shell $1$ is the set of all electrons of an atom with principal quantum number $1$, and subshell $s$ is the set of all electrons of an atom with quantum azimuthal number $0$. Orbital $1s$ is not a place either, it's a probability function that tells you how likely you are to find an electron with $n = 1$ and $\ell=0$ in a given point in space. Still, the plot for $1s$ will basically describe a physical location around the nucleus, if you consider all points in space where the probability isn't $0$ as points of that location. (Or nearly zero, as I seem to recall it's never exactly zero.)

However, I also read that for $k > l$, electrons with $n=k$ are more energetic and hence farther away from the nucleus than electrons with $n=l$. So, I guess that there are only certain regions of space around the nucleus where an electron with $n=k$ can be, and that they're generally larger than the regions described by orbitals.

So, ultimately, I guess that both shells and orbitals can be seen as locations, although they aren't strictly speaking. I don't know about subshells, because I'm not sure if electrons can only have specific values $\ell$ at a specific distance from the nucleus.

Am I making any sense?

Answer 1

Your question is quite a fundamental one.

First, you're right that shells, subshells, and orbitals are not, strictly speaking, single "locations." To think so is to adopt a classical picture of the atom. In such a picture electrons orbit at some fixed radius around the nucleus, and their positions can in principle be known, along with their momenta and angular momenta, using Newton's Second Law and other basic equations of classical mechanics.

But quantum mechanics doesn't work that way. Rather, quantum mechanics is governed by the Schrödinger equation, which does not govern positions, momenta, and so on, but rather wavefunctions $\psi(x)$. An electron's wavefunction, in turn, tells you the probability of finding the electron at a location if you look there. (This probability density is $|\psi(x)|^2$.)

The "plots" of shells and subshells that you refer to are in fact plotting this $|\psi(x)|^2$ quantity, and typically brighter areas are areas of greater probability for the electron to be found. The picture below is taken from Wikipedia's "Hydrogen Atom" article.

For example, the 1s orbital of hydrogen (pictured top left) has a high probability of being found in the middle and a small probability of being found far away from the middle, and is also symmetric about the origin. (That symmetry is true of all s subshells.)

It may seem weird that these wavefunctions are the stable states of the hydrogen atom. I will respond somewhat flippantly that quantum mechanics is weird like that. These states are in fact state of definite energy, meaning if the electron is in one of these states (has one of these wavefunctions), we know exactly what its energy is. If the electron has a different wavefunction, we do not know what its energy is.

The hydrogen wavefunction is completely specified by four quantum numbers:

• Principal quantum number $n = 1,2,3,\dots$
• Orbital angular momentum quantum number $l$ between 0 and $n - 1$ in integer steps. These get special labels, with $l=0$ being labeled $s$, $l=1$ being labeled $p$, $l=2$ being labeled $d$, $l=3$ being labeled $f$, and so on.
• Azimuthal angular momentum quantum number $m_l$ between $-l$ and $+l$ in integer steps.
• Spin quantum number $m_l$, which is always $-1/2$ or $+1/2$.

As you say, "shells" consist of all the states with the same $n$, "subshells" of all states with the same $n$ and $l$, and orbitals of all the states with the same $n$, $l$, and $m_l$. Each orbital can only have two electrons because, once $n,l,m_l$ are all specified, the only remaining quantum number you need to specify the state of the electron is the spin quantum number, which only has two possible values.

All of this comes from the simple application of the Schrödinger equation, which I always thought was really neat.

The various shells and subshells and orbitals have different energies as well as different wavefunctions, which is what you state at the end. There isn't necessarily a neat correspondence between the size and shape of the wavefunction and the corresponding state's energy, but broadly speaking for the hydrogen atom, the larger the wavefunction, the higher the energy. This is because, again broadly speaking, if the electron spends more time further away from the nucleus, then the negative potential energy of the Coulomb interaction between the electron and nucleus is smaller in magnitude, meaning the energy is larger.

The different angular momenta of the electron (orbital and spin) interact to produce more complex energy dependence than the simple dependence on $n$. This is called "fine structure" and "hyperfine structure" and is why $2s$ has a lower energy than $2p$, for example.