Quantum Numbers
The positions of electrons in an atom as well as their energies are described by their wavefunctions. This leads to atomic orbitals, regions of space where electrons are likely to be found (i.e. an electron’s probability density). These orbitals have specific energies and shapes. Only two electrons can be located in any given atomic orbital.
The quantum numbers are a set of numbers (n, l, ml, ms) that fully characterize the possible states of a system, or the state of an electron in the hydrogen atom (or hydrogen-like atom; a particle with one electron).
Principal, n
The principal quantum number, n, indicates orbital size and energy. Larger n means a larger, higher-energy orbital. A 2s orbital is larger than a 1s; a 3p is larger than a 2p.
The principal quantum number is a positive integer:
\[n = 1, ~2, ~3, \ldots\]
The principal quantum number defines the electron shell (or energy level). An electron with n = 1 occupies the first shell; n = 2 is the second shell, and so on. Higher shells correspond to greater average distance from the nucleus and higher energy.
NoteWhy Are Shells Called K, L, M, N…?
You may encounter an older naming system where n = 1 is the “K shell,” n = 2 is the “L shell,” n = 3 is the “M shell,” and so on. This lettering comes from early X-ray spectroscopy.
In 1913, British physicist Henry Moseley studied characteristic X-rays emitted by different elements. He observed distinct series of spectral lines and labeled them K, L, M, starting from the middle of the alphabet. He deliberately avoided A, B, C in case additional series were discovered at higher or lower energies. (They weren’t, but the K, L, M nomenclature stuck.)
When Bohr’s atomic model explained these X-ray lines as transitions between electron shells, the spectroscopic labels transferred to the shells themselves. The K series corresponded to transitions ending at n = 1, the L series to n = 2, and so on.
Today, the numerical quantum number n is standard, but K, L, M, N shells still appear in X-ray spectroscopy, materials science, and some older chemistry texts.
Azimuthal, l
The azimuthal quantum number (or orbital angular momentum quantum number), l, determines the shape of orbitals and defines a subshell. A subshell is a set of orbitals within a shell that share the same n and l values. For example, the 2p subshell contains three orbitals (2px, 2py, 2pz), all with n = 2 and l = 1.
l is an integer between 0 and n − 1:
\[l = 0, ~1, ~2, ~3, \ldots, ~n-1\]
Each shell (n) contains exactly n subshells. For example, the third shell (n = 3) contains three subshells: 3s (l = 0), 3p (l = 1), and 3d (l = 2).
Each subshell type has a characteristic shape and letter label:
Orbitals with l = 0 are called s orbitals; orbitals with l = 1 are called p orbitals, and so on.
s-type orbitals are spherical in shape while p-type orbitals have a “dumbbell” like shape.
Magnetic, ml
The magnetic quantum number, ml, indicates the orientation in space of an orbital within a subshell. In the absence of an external magnetic field, orbitals with different ml values but the same n and l have identical energy. The magnetic quantum number spans from −l to +l:
\[m_{\mathrm{l}} = 0, \pm 1, \pm 2, \ldots, \pm l\]
There are 2l+1 values of ml for a given subshell, l. For example, an electron with a value of l = 1 can have one of three values for ml (–1, 0, 1). That is, an electron in a p orbital can reside in one of the three orientations of that p orbital of shell n.
Note: There is no specific assignment of ml to a particular orientation (we do not assign ml to, say, a px orbital).
Spin, ms
The spin quantum number, ms, indicates the “spin” of an electron. Thus far we have introduced three quantum numbers that localize a position to an orbital of a particular size, shape and orientation. We now introduce the fourth quantum number that describes the type of electron that can be in that orbital. Recall that two electrons can reside inside one atomic orbital.
We can define each one uniquely by indicating the electron’s spin. According to the Pauli-exclusion principle, no two electrons can have the exact same four quantum numbers. This means that two electrons in one atomic orbital cannot have the same “spin”. We generally denote “spin-up” as ms = 1/2 and spin-down as ms = –1/2.
\[m_{\mathrm{s}} = \dfrac{1}{2} ~~\mathrm{or}~~ -\dfrac{1}{2}\]
While it is easy to rationalize the spin of an electron as a “spinning planet” (classical spin), where a counter-clockwise rotation results in a “spin-up” designation while a clockwise rotation results in a “spin-down”, this is not reality. Electrons in atoms are waves and do not behave as a spinning planet. They have a “quantum spin”. Nevertheless, the property of spin arises from experimental observations such as the Stern-Gerlach experiment.
TipWhat do the quantum numbers actually signify?
Quantum numbers give information about the location of an electron or set of electrons. A full set of quantum numbers describes a unique electron for a particular atom.
Think about it as the mailing address to your house. It allows one to pinpoint your exact location out of a set of n locations you could possibly be in. We can narrow the scope of this analogy even further. Consider your daily routine. You may begin your day at your home address but if you have an office job, you can be found at a different address during the work week. Therefore we could say that you can be found in either of these locations depending on the time of day. The same goes for electrons. Electrons reside in atomic orbitals (which are very well defined ‘locations’). When an atom is in the ground state, these electrons will reside in the lowest energy orbitals possible (e.g. 1s2, 2s2, and 2p2 for carbon). We can write out the physical ‘address’ of these electrons in a ground-state configuration using quantum numbers as well as the location(s) of these electrons when in some non-ground (i.e. excited) state.
You could describe your home location any number of ways (GPS coordinates, qualitatively describing your surroundings, etc.) but we’ve adapted to a particular formalism in how we describe it (at least in the case of mailing addresses). The quantum numbers have been laid out in the same way. We could communicate with each other that an electron is “located in the lowest energy, spherical atomic orbital” but it is much easier to say a spin-up electron in the 1s orbital instead. The four quantum numbers allows us to communicate this information numerically without any need for a wordy description.
Of course carbon is not always going to be in the ground state. Given a wavelength of light for example, one can excite carbon in any number of ways. Where will the electron(s) go? Regardless of what wavelength of light we use, we know that we can describe the final location(s) using the four quantum numbers. You can do this by writing out all the possible permutations of the four quantum numbers. Of course, with a little more effort, you could predict the exact location where the electron goes but in my example above, you know for a fact you could describe it using the quantum number formalism.
The quantum numbers also come with a set of restrictions which inherently gives you useful information about where electrons will NOT be. For instance, you could never have the following possible quantum numbers for an atom:
n = 1; l = 0; ml = 0; ms = 1/2
n = 1; l = 0; ml = 0; ms = –1/2
n = 1; l = 0; ml = 0; ms = 1/2
This set of quantum numbers indicates that three electrons reside in the 1s orbital which is impossible!
As Jan stated in his post, these quantum numbers are derived from solutions to the Schrödinger equation for the hydrogen atom (or any one-electron system). The solutions correspond to the possible energy levels of the hydrogen atom. Remember, energy is QUANTIZED (as postulated by Max Planck). That means that an energy level may exist (arbitrarily) at 0 and 1 but NEVER in between. There is a discrete ‘jump’ in energy levels and not some gradient between them. From these solutions a formalism was constructed to communicate the solutions in a very easy, numerical way just as mailing addresses are purposefully formatted in such a way that is easy that anyone can understand with minimal effort.
In summary, the quantum numbers not only tell you where electrons will be (ground state) and can be (excited state), but also will tell you where electrons cannot be in an atom (due to the restrictions for each quantum number).
Source: Chemistry StackExchange.
Practice
For the third shell (n = 3), determine:
- all possible values of l
- all possible values of ml for each subshell
- the total number of orbitals in the shell
- the maximum number of electrons that can occupy this shell
Solution
Part a: Possible values of l
Since l ranges from 0 to n − 1:
\[l = 0, ~1, ~2\]
This gives three subshells: 3s (l = 0), 3p (l = 1), and 3d (l = 2).
Part b: Possible values of ml for each subshell
Since ml ranges from −l to +l:
- For l = 0 (3s): ml = 0 only → 1 orbital
- For l = 1 (3p): ml = −1, 0, +1 → 3 orbitals
- For l = 2 (3d): ml = −2, −1, 0, +1, +2 → 5 orbitals
Part c: Total orbitals in the shell
\[1 + 3 + 5 = 9~\text{orbitals}\]
In general, shell n contains n2 orbitals: 32 = 9 ✓
Part d: Maximum electrons in the shell
Each orbital holds at most 2 electrons (ms = +½ or −½):
\[9~\text{orbitals} \times 2~\text{electrons/orbital} = 18~\text{electrons}\]
In general, shell n holds at most 2n2 electrons: 2(3)2 = 18 ✓
What’s Next
Quantum numbers define which orbitals exist, but what do these orbitals actually look like? The Atomic Orbitals page covers orbital shapes, sizes, and probability distributions (the three-dimensional regions where electrons are likely to be found). You’ll need these shapes to understand chemical bonding and molecular geometry.
Once you know the orbitals, the question becomes: how do electrons fill them? The Electron Configurations page covers the filling rules and explains why those arrangements determine an element’s chemistry.