Energy Levels and Quantum Numbers

A periodic table is pinned to the back wall of every Class-11 chemistry classroom in India. It has a particular shape — two columns on the left, a block of ten columns in the middle, six columns on the right, and a separate two-row strip of fourteen elements at the bottom. If you asked your chemistry teacher why the table has exactly that shape — why exactly 2 in the first row, 8 in the second, 8 in the third, 18 in the fourth — you probably got an answer that used the words "shells" and "subshells" without telling you where the numbers 2, 8, 18 come from.

They come from quantum mechanics. More specifically, they come from a set of four integers called quantum numbers and one rule called the Pauli exclusion principle. Together those five things determine, with no adjustable parameters, that hydrogen has one electron, helium has two, lithium has three, that helium is inert and lithium is explosive, that the second row holds 8 elements and the fourth holds 18, and that iron has unpaired electrons and is magnetic. Every question you were ever told to "just memorise" in school chemistry has an answer in the quantum numbers.

This article gives them to you. Start with where Bohr's n came from, see why it is not enough, and work out how three more numbers — \ell, m_\ell, m_s — fall out of the full quantum-mechanical analysis.

What Bohr missed

The Bohr model quantised angular momentum via L = n\hbar, producing the ladder of energies

It fit the hydrogen spectrum beautifully. But even for hydrogen, two observations told physicists something was missing.

Observation 1: spectral lines are split. Look at the Balmer H-\alpha line with a high-resolution spectrograph. What looked like a single line at 656.3 nm is actually two lines, separated by about 0.016 nm. The single n = 3 level is not one state — it is several states with slightly different energies. Bohr's picture had no way to produce this splitting.

Observation 2: magnetic field splitting (Zeeman effect). Put a discharge tube in a magnetic field. Each spectral line splits into several lines whose separation is proportional to the field strength. In Bohr's picture, a single circular orbit can point in any direction in space, so all orientations should have the same energy — no splitting. But the observed splitting is discrete: the line splits into three (or five, or seven) components, never a continuous spread.

Both observations point to the same conclusion: the n-th Bohr level is not a single state. It is a bundle of several states, which happen to have (almost) the same energy in the absence of external fields but reveal themselves under high resolution or external perturbation.

The full answer came from solving the Schrödinger equation for the hydrogen atom. The result: for each n there are not one but 2n^2 independent single-electron states, labelled by three more integers in addition to n. Those three numbers are \ell, m_\ell, and m_s.

The principal quantum number n

The first of the four quantum numbers is the old friend from Bohr.

n sets the energy of the level for a hydrogen-like atom:

And n sets the average distance of the electron from the nucleus. In the full quantum picture, an electron does not orbit at a fixed radius; it is a cloud with a probability distribution. But the mean radius scales exactly as Bohr predicted:

Shell is the old chemistry name for a fixed n. The n = 1 shell is called the K-shell, n = 2 the L-shell, n = 3 the M-shell, and so on. The letter sequence K, L, M, N, O, \ldots is a historical convention from X-ray spectroscopy.

The angular momentum quantum number \ell

Bohr assumed the electron orbits on a flat circle. The Schrödinger equation does not — it is a 3D differential equation, and solutions in 3D come in many more flavours than circles.

In 3D, angular momentum is a vector: \vec L = \vec r \times \vec p. Its magnitude squared |\vec L|^2 and its z-component L_z are separately quantised, and the quantisation rules are:

The integer \ell is the orbital angular momentum quantum number (or "azimuthal quantum number" in old textbooks). For a given n, the allowed values are

Why this range: the solution to the radial Schrödinger equation for hydrogen only gives a normalisable wavefunction when \ell < n. There is a finite number of angular "modes" that fit inside the n-th radial mode. This is analogous to how a drum head of a given radius supports only a finite number of vibrational patterns at each radial frequency.

So for n = 1: only \ell = 0. For n = 2: \ell = 0 and \ell = 1. For n = 3: \ell = 0, 1, 2. For n = 4: \ell = 0, 1, 2, 3. And so on.

Each value of \ell has a traditional letter name from the early spectroscopy literature (the letters are short for "sharp", "principal", "diffuse", "fundamental" — properties of the spectral lines). After f, it just goes alphabetically.

A combination of n and \ell is called a subshell. The n = 2, \ell = 0 subshell is "2s". The n = 3, \ell = 1 subshell is "3p". The combinations you will see in school chemistry: 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f — these are the fundamental categories into which electrons are organised.

The magnetic quantum number m_\ell

For a given \ell, the angular momentum vector \vec L can point in different directions. But not any direction — the z-component is quantised:

There are 2\ell + 1 allowed values of m_\ell.

Why this range: the wavefunction must be single-valued as you walk around the z-axis. This boundary condition forces the angular factor to be of the form e^{im_\ell\phi} with m_\ell an integer. And |L_z| cannot exceed the magnitude of \vec L, so |m_\ell| \leq \ell.

For an s-orbital (\ell = 0): only m_\ell = 0. Just one orientation (trivially — a sphere has no orientation).

For a p-orbital (\ell = 1): m_\ell = -1, 0, +1. Three distinct orientations — the p_x, p_y, p_z orbitals you learn in chemistry.

For a d-orbital (\ell = 2): m_\ell = -2, -1, 0, +1, +2. Five orientations — d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, d_{z^2} in the chemistry notation.

For an f-orbital (\ell = 3): seven orientations.

The reason m_\ell is called the "magnetic" quantum number: in a magnetic field \vec B along z, the energy of an orbital shifts by

where \mu_B = e\hbar/(2m_e) = 9.274\times 10^{-24} J/T is the Bohr magneton. Different m_\ell values shift by different amounts — the Zeeman effect. The quantum number m_\ell was invisible until the Zeeman measurement forced it out.

The spin quantum number m_s

Here is an observation that the Schrödinger equation for a three-dimensional charged particle orbiting a proton does not explain: even with zero orbital angular momentum (\ell = 0), an electron has an intrinsic magnetic moment of magnitude \mu_B. Stern and Gerlach in 1922 showed this directly — a beam of silver atoms (whose only valence electron is in a 5s orbital, so \ell = 0) passed through a non-uniform magnetic field split into two beams, not one. The electron carries its own angular momentum, independent of orbital motion.

The extra angular momentum is called spin. It is quantised like the orbital angular momentum, with a total-spin quantum number s = 1/2 and projection quantum number

The corresponding z-component of spin angular momentum is S_z = m_s\hbar = \pm\hbar/2.

Spin is a fundamentally quantum property. It is not an electron "spinning on its axis" — treating an electron as a classical sphere spinning fast enough to produce \hbar/2 of angular momentum requires surface speeds greater than the speed of light. Spin is intrinsic to the particle, like charge. You do not derive it from classical pictures; it falls out of the Dirac equation (the relativistic quantum equation of the electron). In non-relativistic quantum mechanics you simply postulate it.

An electron with m_s = +1/2 is called "spin up"; with m_s = -1/2, "spin down". A conventional arrow notation is ↑ and ↓.

Adding spin doubles the number of states: for every (n, \ell, m_\ell) combination, there are two spin states. So in total, an atom has 2n^2 single-electron states per shell.

Those numbers — 2, 8, 18, 32 — are exactly the lengths of the rows of the periodic table (with some rearrangement due to which subshells are at what energy). The periodic table is, at heart, a list of the quantum states of a hydrogen-like atom, filled one electron at a time.

Explore the quantum numbers

Drag the three sliders below to pick (n, \ell, m_\ell). The allowed ranges update automatically — \ell cannot exceed n - 1, m_\ell cannot exceed \ell in magnitude. Read off the shape (s/p/d/f) and the number of states in the current subshell.

The Pauli exclusion principle

Here is the rule that ties everything together. Wolfgang Pauli proposed it in 1925:

No two electrons in the same atom can have the same set of four quantum numbers (n, \ell, m_\ell, m_s).

Equivalently: every orbital (n, \ell, m_\ell) can hold at most two electrons, and they must have opposite spins.

Why: the deep reason is that electrons are fermions — their wavefunction must be antisymmetric under exchange of two particles, which forces the two-electron wavefunction to vanish if they are in the same single-particle state. This is a theorem (the spin-statistics theorem) in relativistic quantum field theory. In school physics, you take the Pauli principle as a postulate; in the full theory, it is derived.

The Pauli principle turns a list of available quantum states into a filling order. Start with the one-electron ground state (n = 1, \ell = 0, m_\ell = 0, m_s = +1/2). Add a second electron: n = 1, \ell = 0, m_\ell = 0, m_s = -1/2. Now the 1s orbital is full. A third electron cannot go into 1s (all four slots — actually just the two spin slots — are taken). It must go into the next available orbital, which is 2s.

This is the Aufbau principle ("aufbau" is German for "building up"): fill the lowest-energy orbitals first, subject to Pauli. The rough ordering of subshell energies (for neutral atoms) is

Notice that 4s is below 3d (which is why potassium and calcium fill 4s before 3d), and 5s below 4d. This ordering is called the Madelung rule and comes from the way shielding by inner electrons shifts the energies from the pure -13.6/n^2 pattern.

Building the periodic table, one element at a time

• Hydrogen (Z = 1): 1 electron. Configuration 1s^1. One electron in 1s.
• Helium (Z = 2): 2 electrons. Configuration 1s^2. Both spin slots of 1s are filled. The n = 1 shell is complete. Helium is inert because it has a complete shell.
• Lithium (Z = 3): 1s^2 2s^1. The third electron is forced into the much more loosely bound 2s. Lithium has one unpaired electron outside a closed shell, which is why it is highly reactive — that outer electron is easy to give up.
• Carbon (Z = 6): 1s^2 2s^2 2p^2. The two 2p electrons go into different m_\ell orbitals (Hund's rule — see below) with parallel spins.
• Neon (Z = 10): 1s^2 2s^2 2p^6. The n = 2 shell is complete. Neon is inert.
• Sodium (Z = 11): 1s^2 2s^2 2p^6 3s^1. One electron outside a full shell — and sodium is explosively reactive, for the same reason lithium is.

Follow this pattern all the way up and you generate the entire periodic table. The length of each row (2, 8, 8, 18, 18, 32) is the number of electrons needed to fill all subshells whose energies lie between two consecutive ns fillings. Every column is a family of atoms with the same outer-shell electron configuration — and therefore similar chemistry.

Hund's rule

When filling a subshell with several degenerate orbitals (like the three p-orbitals, all at the same energy in an isolated atom), Hund's rule says electrons first occupy different orbitals with parallel spins, and only pair up after each orbital has one electron. Iron (Z = 26) has the configuration [Ar]\ 3d^6\ 4s^2. Of the six 3d electrons, five fill all five 3d orbitals with parallel spins; the sixth must pair up. That leaves 4 unpaired electrons in iron, which is why iron is ferromagnetic: those unpaired spins line up with each other inside iron's crystal lattice, producing macroscopic magnetisation. The magnets stuck to your fridge in Jaipur or Bengaluru work because of the Pauli principle plus Hund's rule.

Spin and NMR — real-world chemistry

Spin is not an abstraction — it is measured directly, every day, in NMR (Nuclear Magnetic Resonance) machines at AIIMS hospitals and in university chemistry departments throughout India. The physics is:

• A proton (like an electron) has spin 1/2, so m_s = \pm 1/2 with S_z = \pm\hbar/2.
• Place a proton in a strong magnetic field B (typically 1.5 T in medical scanners). The two spin states split in energy by \Delta E = g_p\mu_N B, where \mu_N = e\hbar/(2m_p) is the nuclear magneton and g_p \approx 5.586 is the proton g-factor.
• Shine radio-frequency electromagnetic radiation of frequency f = \Delta E/h at the sample. Protons in the low state absorb and flip to the high state. The absorption is detected — that is the NMR signal.

The splitting is tiny compared to atomic transitions (MHz, not eV), but readily detectable. Chemists use it to identify functional groups in organic molecules; doctors use it (as MRI) to image soft tissues. Every MRI scan you see on a hospital noticeboard is a map of the density and environment of proton spins in water and fat. It works only because protons have spin quantised as m_s = \pm 1/2.

Worked examples

Common confusions

• "Electrons literally spin on their axes." No. Spin is intrinsic angular momentum — a quantum property of the electron that behaves mathematically like angular momentum but has no classical "spinning" interpretation. Don't picture a tiny rotating ball. Picture a label that says "+1/2" or "−1/2" attached to each electron.

• "Shell, subshell, and orbital are the same thing." They are not. A shell is all states with the same n. A subshell is all states with the same (n, \ell). An orbital is one specific (n, \ell, m_\ell) — a single spatial wavefunction that can hold up to 2 electrons (opposite spins). Example: n = 2 is one shell, which contains two subshells (2s and 2p), which contain 4 orbitals in total (2s: 1, 2p: 3).

• "Why is \ell = n not allowed?" The radial Schrödinger equation has normalisable solutions only when the radial quantum number n_r = n - \ell - 1 \geq 0, i.e. \ell \leq n - 1. Setting \ell = n would require a radial wavefunction with no oscillations and no decay — mathematically, the only solution to that boundary condition is the zero function, which isn't a physical state.

• "Pauli exclusion only applies to electrons." It applies to all fermions — half-integer-spin particles. Protons, neutrons, and electrons are all fermions. The reason neutron stars are stable against gravitational collapse is Pauli exclusion among neutrons: two neutrons cannot occupy the same quantum state, and the resulting "degeneracy pressure" supports the star against its own gravity. Bosons (integer spin — photons, gluons, and helium-4 atoms under the right conditions) do not obey Pauli exclusion and can pile up arbitrarily many into the same quantum state — which is why you can have things like Bose-Einstein condensates and laser beams.

• "The four quantum numbers are specific to hydrogen." The rules \ell \leq n-1, |m_\ell| \leq \ell, m_s = \pm 1/2 are specific to one electron in a central potential. For multi-electron atoms, those labels are approximate — the true energy eigenstates are combinations of products of single-electron orbitals, and strictly speaking neither \ell nor m_\ell for individual electrons is exactly conserved. But the approximation is so good (for ground states and low excitations) that chemists work with these labels every day. JEE and NCERT use them exclusively.

• "Spin is magic." It is not magic; it is required. Once you demand a quantum theory consistent with special relativity, Dirac's equation falls out — and solutions to Dirac's equation for a single free particle naturally have four components, two of which correspond to particles (electron) of spin up and spin down, and the other two to antiparticles (positron). Spin is the shadow in non-relativistic quantum mechanics of a structure that is built into relativistic quantum mechanics from the start.

Where this leads next

• Hydrogen spectrum and spectral series — the transitions between levels labelled by n and the wavelengths they emit.
• Bohr's model of the hydrogen atom — the first successful quantisation of the hydrogen atom, which gave only the principal quantum number.
• de Broglie hypothesis — the matter-wave picture that makes the standing-wave quantisation conditions natural.
• X-rays — production and properties — transitions involving inner-shell (K, L) electrons in heavy atoms, whose energies scale as Z^2.
• The nucleus — composition and size — the next layer of structure, where quantum numbers again classify proton and neutron states.
• Photoelectric effect — photon absorption by bound electrons, where the full set of quantum numbers determines selection rules.