Quantum numbers and atomic orbitals

Result

Quantum numbers and atomic orbitals

T-003Home CU-101Threads quantum · structure

Statement

Solving the time-independent Schrödinger equation for an electron in a hydrogen-like atom, by separating variables in spherical coordinates, forces every stationary electron state to be labelled by three integers arising from the boundary conditions on the separated equation, plus a fourth from electron spin: the principal quantum number \(n=1,2,3,\dots\) (energy shell), the orbital angular momentum quantum number \(l=0,1,\dots,n-1\) (orbital shape), the magnetic quantum number \(m_l=-l,\dots,+l\) (orbital orientation), and the spin quantum number \(m_s=\pm\tfrac12\) (intrinsic electron spin). Each allowed \((n,l,m_l)\) triple defines one atomic orbital — a specific spatial probability-density pattern — and each orbital holds at most two electrons, one with each value of \(m_s\).

Why it matters

This result replaces the Bohr model's single quantum number \(n\) with the full structure needed to explain everything the Bohr model could not: why atomic spectra split into fine sub-lines, why the periodic table has the block structure it does (s, p, d, f blocks track directly onto \(l=0,1,2,3\)), and why atoms have a definite, finite size rather than collapsing — the Pauli exclusion principle, built directly on these four labels, is what forces electrons to stack into successively higher shells instead of all piling into the lowest-energy orbital.

It is also the last piece of machinery this unit needs before building up real, multi-electron atoms: the building-up principle, Hund's rule, and every periodic trend in the rest of CU-101 are all statements about how electrons distribute themselves across the \((n,l,m_l,m_s)\) labels introduced here.

Hypotheses

The potential is spherically symmetric (Coulombic, depending only on \(r=|\mathbf r|\)).Separation of the Schrödinger equation into radial and angular parts (Step 2 below) relies on the potential energy term containing no angular dependence. In a molecule, where an electron feels the pull of two or more nuclei at fixed positions, the potential is not spherically symmetric about any single centre, and the clean separation into radial \(\times\) angular factors fails — molecular orbitals are built from linear combinations of atomic orbitals precisely because the atomic \((n,l,m_l)\) labels no longer solve the molecular problem outright. Single electron, no external field (for the exact hydrogen-like solution).With a magnetic field present, the energy additionally splits by \(m_l\) (and separately by \(m_s\)) — the Zeeman effect — because the field breaks the rotational symmetry that made \(m_l\) states degenerate in the field-free case. With more than one electron, electron–electron repulsion breaks the exact \((n,l)\)-degeneracy that holds for hydrogen (see Discussion), though the \((n,l,m_l,m_s)\) labelling scheme itself survives as an approximate, and extremely useful, orbital picture.

Proof

-\frac{\hbar^2}{2m_e}\nabla^2\psi - \frac{Ze^2}{4\pi\varepsilon_0r}\psi = E\psi

The time-independent Schrödinger equation for a single electron in the Coulomb field of a nucleus of charge \(+Ze\); \(\psi(r,\theta,\phi)\) is the electron's wavefunction. B

\psi(r,\theta,\phi) = R(r)\,Y(\theta,\phi) \quad\Longrightarrow\quad \frac{1}{R}\frac{d}{dr}\!\left(r^2\frac{dR}{dr}\right) - \frac{2m_er^2}{\hbar^2}\!\left(\frac{Ze^2}{4\pi\varepsilon_0r}+E\right) = \frac{1}{Y}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial Y}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2Y}{\partial\phi^2}\right]

Separation of variables (the general method proved in this network's maths2u reference, applied here to a spherically symmetric potential): substitute the product ansatz into Step 1 and divide through by \(RY\), isolating all \(r\)-dependence on the left and all \((\theta,\phi)\)-dependence on the right. As in every separation-of-variables argument, both sides must equal the same constant. B

\text{Angular equation: eigenvalue } -l(l+1),\ \ l=0,1,2,\dots,\qquad Y_l^{m_l}(\theta,\phi),\ \ m_l=-l,\dots,+l

Named lemma (spherical harmonics, cited not re-derived here): the angular equation from Step 2, on its own, is a well-studied eigenvalue problem whose only solutions single-valued and finite over the full sphere are the spherical harmonics \(Y_l^{m_l}\), which exist precisely for non-negative integer \(l\) and integer \(m_l\) with \(|m_l|\le l\) — forcing \(2l+1\) values of \(m_l\) for each \(l\). This is where \(l\) and \(m_l\) originate: as the two integers indexing which angular eigenfunction the electron occupies. C

\text{Radial equation with } l(l+1)\text{ fixed: bound-state solutions } R_{n,l}(r) \text{ exist only for } n=l+1,\,l+2,\,l+3,\dots

Named lemma (radial hydrogen equation, cited not re-derived here): substituting the angular eigenvalue \(-l(l+1)\) from Step 3 into the radial equation of Step 2 and demanding a normalisable (finite total probability), bound (\(E<0\)) solution forces the energy to take one of a discrete set of values, indexed by an integer \(n>l\); relabelling \(n\) as starting from \(1\) (so \(n\ge l+1\), equivalently \(l\le n-1\)) recovers exactly the principal quantum number, with \(E_n=-13.6Z^2/n^2\,\text{eV}\) — the same formula as the Bohr model, but now derived from the full wave equation rather than postulated orbits. C

S_z\,\chi_\pm = \pm\tfrac12\hbar\,\chi_\pm \qquad (\text{spin: a separate, non-orbital degree of freedom})

Electron spin does not arise from Steps 1–4 at all — it is not a solution of the Schrödinger equation, which contains no spin term. It is an independent postulate (confirmed experimentally by the Stern–Gerlach 1922 silver-atom deflection experiment, and derived from first principles only in the relativistic Dirac equation, 1928) that every electron carries an intrinsic angular momentum with exactly two possible \(z\)-projections, labelled \(m_s=+\tfrac12\) and \(m_s=-\tfrac12\). B

\text{Each state} \ (n,l,m_l,m_s) \text{ is a distinct quantum state; an orbital} \ (n,l,m_l) \text{ holds at most 2 electrons.}

Combine Steps 3–5: the full description of a one-electron state needs all four labels. Fixing \((n,l,m_l)\) but leaving \(m_s\) free defines one atomic orbital (a fixed spatial wavefunction), and the Pauli exclusion principle (cited, not proved here — it is a separate postulate about indistinguishable fermions) forbids two electrons from sharing all four labels, capping any one orbital's occupancy at the two spin states. A

Result

n=1,2,3,\dots;\quad l=0,\dots,n-1;\quad m_l=-l,\dots,+l;\quad m_s=\pm\tfrac12

Reading. \(n\) sets the energy shell; \(l\) sets the orbital's shape and is conventionally relabelled \(s,p,d,f,\dots\) for \(l=0,1,2,3,\dots\); \(m_l\) sets the orbital's spatial orientation among the \(2l+1\) options at that \(l\); \(m_s\) is the two-valued spin label that lets two electrons share one spatial orbital.

Scope. Exact for any one-electron atom or ion. In multi-electron atoms the labels \((n,l,m_l,m_s)\) remain a valid, standard way to classify electron states (via the orbital approximation), but the energies they correspond to depend on both \(n\) and \(l\), unlike the pure hydrogen case, where energy depends on \(n\) alone (see Discussion).

Corollaries & converses

The number of orbitals in a shell of principal number \(n\) is \(\sum_{l=0}^{n-1}(2l+1)=n^2\), and the maximum electron capacity of that shell is \(2n^2\) — the origin of the familiar \(2,8,18,32,\dots\) shell-filling sequence.
The letters \(s,p,d,f\) (for \(l=0,1,2,3\)) are historical spectroscopic labels (sharp, principal, diffuse, fundamental) predating the quantum-mechanical explanation of what they label — a naming fossil from 19th-century spectroscopy, now permanently attached to the orbital-shape quantum number.
Converse: given any set of four values obeying the stated ranges, Steps 3–5 guarantee a corresponding physical state exists; the ranges are not just necessary but sufficient conditions for a valid one-electron quantum state.

Fails without

Drop spherical symmetry (e.g. a diatomic molecule): in H\(_2^+\) (one electron, two protons), the potential energy depends on the distance to each proton separately, not on a single radial coordinate \(r\), so the separation of Step 2 does not apply as written; the correct molecular orbitals (\(\sigma,\pi,\dots\)) are built by combining atomic \(s,p,\dots\) orbitals rather than solving a fresh spherically-symmetric problem.
Ignore the Pauli restriction on \(m_s\) (Step 6): without it, nothing prevents every electron in an atom from occupying the lowest-energy \(1s\) orbital; real matter (and the entire periodic table's shell structure) depends on the exclusion principle forcing electrons to stack upward through successive shells, which is exactly what gives atoms their characteristic sizes and chemical valence.

Common errors

Writing \(l=n\) instead of \(l\le n-1\) — a common slip that would (wrongly) allow, for instance, a \(1p\) orbital (\(n=1,l=1\)), which does not exist; the largest \(l\) available in shell \(n\) is \(n-1\).
Treating \(m_l\) and \(m_s\) as measuring the same thing. \(m_l\) is the projection of orbital angular momentum (from the electron's spatial motion); \(m_s\) is the projection of an entirely separate, intrinsic spin angular momentum that has no classical spatial-motion analogue.
Assuming energy in a multi-electron atom depends on \(n\) alone, as it does for hydrogen. In real multi-electron atoms, \(4s\) can be lower in energy than \(3d\) (the basis of several "anomalous" ground-state configurations covered in this unit's building-up-principle result) precisely because \(l\), not just \(n\), affects the energy once electron–electron repulsion is present.

Discussion

Bohr's 1913 model needed only one quantum number because it assumed a definite circular orbit; as spectroscopists found ever finer splitting in atomic spectra through the 1910s and 1920s (fine structure, the Zeeman effect, the anomalous Zeeman effect), first Sommerfeld generalised Bohr's circular orbits to ellipses (introducing a second number, essentially an early form of \(l\)), and separately, the puzzling factor of two in certain spectral splittings led Uhlenbeck and Goudsmit to propose electron spin in 1925 — initially as a literal spinning charged sphere, an image immediately known to be physically untenable (it would require the electron's surface to move faster than light), yet the underlying two-valued quantum number it introduced turned out to be exactly right.

The full, correct four-number picture only became rigorous with Schrödinger's 1926 wave equation (giving \(n,l,m_l\) as the natural outcome of solving a differential equation, rather than postulated by hand) and Dirac's 1928 relativistic equation (deriving spin from first principles, as an unavoidable consequence of combining quantum mechanics with special relativity, rather than requiring it as a separate assumption).

The exact \((n,l)\)-independence of hydrogen's energy — every orbital with the same \(n\) has the same energy regardless of \(l\) — is again the "accidental" \(SO(4)\) symmetry of the pure \(1/r\) Coulomb potential (see this unit's Bohr-model page, Discussion). Any deviation from a pure \(1/r\) potential, including the electron-screening effects present in every atom with more than one electron, breaks this extra symmetry and splits the energy by \(l\) as well as \(n\) — the physical origin of the "\(4s\) fills before \(3d\)" anomalies explored in the next result.

Common misconception: that an "orbital" is a path the electron travels along, in the sense of the old Bohr orbit. It is not a trajectory at all; \(\psi_{n,l,m_l}\) is a stationary probability-amplitude cloud, and \(|\psi|^2\) gives the probability density of finding the electron at a given point, with no implication of continuous motion along any particular curve — the word "orbital" was deliberately chosen to distinguish it from "orbit."

Worked examples

n=3: \quad l=0,1,2 \ \ (3s,3p,3d), \qquad \text{orbitals} = 3^2=9, \qquad \text{electron capacity}=2(3^2)=18

List every allowed \(l\) for \(n=3\) (\(l=0,\dots,n-1=0,1,2\)), name the corresponding subshells, and total the orbitals and electron capacity via the corollary formulas. A

l=1\ (p\text{ subshell}): \quad m_l=-1,0,+1 \ \Longrightarrow\ 3\ p\text{ orbitals } (p_x,p_y,p_z), \text{ each holding 2 electrons } \Longrightarrow 6\ p\text{ electrons max}

Any \(p\) subshell, regardless of \(n\), has exactly \(2(1)+1=3\) orbitals from the \(m_l\) range, each capable of holding \(2\) electrons by Step 6, giving the familiar "\(p^6\)" maximum seen throughout the periodic table's \(p\)-block. A

\text{subshell capacities: } s^2,\ p^6,\ d^{10},\ f^{14}\ \ (\text{from } 2(2l+1)\text{ for } l=0,1,2,3)

Reading. The periodic table's block widths (2 columns for \(s\), 6 for \(p\), 10 for \(d\), 14 for \(f\)) are a direct readout of \(2(2l+1)\) for each \(l\) — the periodic table is, in this precise sense, a picture of these four quantum numbers.

Scope. Holds for every element; which subshells are actually being filled, and in what order, is the subject of this unit's building-up-principle result.

Problems

List all allowed \((n,l,m_l)\) combinations for \(n=2\), and state how many total orbitals and how many electrons the \(n=2\) shell can hold.
Solution
\(n=2\Rightarrow l=0,1\). For \(l=0\): \(m_l=0\) (one \(2s\) orbital). For \(l=1\): \(m_l=-1,0,+1\) (three \(2p\) orbitals). Total orbitals: \(1+3=4=n^2=2^2\), matching the corollary. Electron capacity: \(2\times4=8=2n^2\), matching \(2,8,18,\dots\) shell filling.
Explain why a \(2d\) orbital does not exist, using the quantum-number ranges established in the Result.
Solution
The Result requires \(l\le n-1\). For \(n=2\), the maximum allowed \(l\) is \(2-1=1\) (the \(p\) subshell); \(d\) corresponds to \(l=2\), which requires \(n\ge3\). A "\(2d\)" label describes a combination that Step 4 (the radial equation's bound-state condition) simply has no solution for — it is not merely unfilled, it does not exist as a quantum state.
A student writes the electron configuration of a hypothetical atom's outer electron as having quantum numbers \(n=3,\ l=3,\ m_l=0,\ m_s=+\tfrac12\). Identify every error.
Solution
Two independent errors. First, \(l\) must satisfy \(l\le n-1=2\), so \(l=3\) is not allowed for \(n=3\) (the largest allowed \(l\) at \(n=3\) is \(l=2\), the \(3d\) subshell). Second, even taken at face value, \(l=3\) would correspond to an \(f\) orbital, which by the general rule \(n\ge l+1\) first appears at \(n=4\) (\(4f\)), not \(n=3\) — so no correction to \(m_l\) or \(m_s\) alone can fix this state; \(n\) and \(l\) are jointly inconsistent.
Using the corollary \(2n^2\), find the smallest shell \(n\) that can hold at least 40 electrons, and check your answer against the actual orbital count.
Solution
Need \(2n^2\ge40\Rightarrow n^2\ge20\Rightarrow n\ge\sqrt{20}\approx4.47\), so the smallest integer is \(n=5\), giving capacity \(2(5)^2=50\ge40\) (while \(n=4\) gives only \(2(4)^2=32<40\)). Checking directly: for \(n=5\), \(l=0,1,2,3,4\) give \(1+3+5+7+9=25=5^2\) orbitals, and \(2\times25=50\) electrons — confirms the shell-count formula.