Quantum Numbers
~6 min read · Tier toggle changes depth (T1 plain · T2 standard · T3 rigour)
You should be comfortable with the concept of a wavefunction and understand that solving the Schrödinger equation for a hydrogen-like atom yields a set of allowed energy states. Familiarity with basic spherical coordinates is helpful but not strictly required.
Why this matters
Quantum numbers are the address system for electrons in atoms.
Without a rigorous way to label electron states, we could not explain the structure of the periodic table, the rules of electron configuration, or the selection rules governing spectral transitions. The quantum numbers emerge directly from the mathematical boundary conditions of the Schrödinger equation, providing a complete and unique description of each electron's state in a multi-electron atom. This lecture bridges the gap between the abstract wavefunction and the concrete orbital diagrams used in chemical bonding.
The Address System for Electrons
Imagine a large apartment building. To find a specific resident, you need a floor number, a room number, and a seat within that room. Similarly, an electron in an atom is not a tiny planet orbiting a nucleus, but a standing wave spread out in space. To describe exactly where this wave is and how it behaves, we need a set of labels. These labels are the quantum numbers. They are not arbitrary; they are forced upon us by the requirement that the wavefunction must be physically sensible (finite, single-valued, and continuous). The principal quantum number tells us the energy level or "shell." The angular momentum quantum number tells us the shape of the orbital. The magnetic quantum number tells us the orientation of that shape in space. Finally, the spin quantum number describes an intrinsic property of the electron itself, independent of its position. Together, these four numbers provide a unique "address" for every electron in an atom.
Principal and Angular Momentum Quantum Numbers
The principal quantum number, \(n\), determines the overall size and energy of the orbital. It can take any positive integer value: \(n=1, 2, 3, \dots\). As \(n\) increases, the orbital becomes larger and the electron is, on average, further from the nucleus. For hydrogen-like atoms, the energy depends solely on \(n\). However, in multi-electron atoms, electron-electron repulsion causes the energy to also depend on the shape of the orbital.
The orbital angular momentum quantum number, \(l\), defines the shape of the orbital. For a given \(n\), \(l\) can take integer values from \(0\) up to \(n-1\). We use letters to denote these values: \(l=0\) is an s orbital, \(l=1\) is a p orbital, \(l=2\) is a d orbital, and \(l=3\) is an f orbital. The restriction \(l \lt n\) comes from the radial equation: only then does the radial wavefunction stay finite and normalisable. For example, if \(n=1\), the only allowed value is \(l=0\), meaning the first shell contains only an s orbital. If \(n=2\), \(l\) can be \(0\) or \(1\), so the second shell contains s and p orbitals.
Magnetic Quantum Number and Orientation
The magnetic quantum number, \(m_l\), specifies the orientation of the orbital in three-dimensional space. For a given value of \(l\), \(m_l\) can take integer values ranging from \(-l\) to \(+l\). This includes zero. The number of possible values for \(m_l\) is \(2l+1\). This corresponds to the number of distinct orientations an orbital of a given shape can have. For instance, if \(l=1\) (a p orbital), \(m_l\) can be \(-1, 0, +1\). These correspond to the three p orbitals aligned along the x, y, and z axes (though the exact axis assignment is a matter of convention). If \(l=2\) (a d orbital), \(m_l\) can be \(-2, -1, 0, +1, +2\), giving five distinct d orbitals. In the absence of an external magnetic field, these orientations are degenerate, meaning they have the same energy.
Spin Quantum Number and the Pauli Exclusion Principle
The spin quantum number, \(m_s\), describes the intrinsic angular momentum of the electron. Unlike the first three numbers, which arise from the spatial part of the wavefunction, spin is a relativistic quantum mechanical property. An electron can have one of two spin states: \(m_s = +\frac{1}{2}\) (often called "spin up") or \(m_s = -\frac{1}{2}\) (often called "spin down"). This property is crucial because of the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of all four quantum numbers. Since an orbital is defined by \((n, l, m_l)\), it can hold a maximum of two electrons, one with spin up and one with spin down. This principle is the fundamental reason for the structure of the periodic table and the existence of matter as we know it.
The result
Quantum numbers and atomic orbitals
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 \(m_s\).
A common misconception is that the magnetic quantum number \(m_l\) determines the energy of the orbital in a multi-electron atom. In fact, in the absence of an external magnetic field, all orbitals with the same \(n\) and \(l\) (i.e., the same subshell) are degenerate. The energy splitting between different \(l\) values within the same \(n\) shell (e.g., 2s vs 2p) is due to electron shielding and penetration effects, not the orientation defined by \(m_l\). The \(m_l\) value only becomes energetically distinct in the presence of an external magnetic field (Zeeman effect). C
Worked example
Problem. For the fourth shell (\(n = 4\)), list the subshells, count the orbitals, find the maximum number of electrons, and say how many of those electrons can have \(m_l = 0\).
- Allowed \(l\): \(0, 1, 2, 3\), i.e. the 4s, 4p, 4d and 4f subshells.
- Orbitals per subshell, \(2l + 1\): \(1 + 3 + 5 + 7 = 16 = n^2\).
- Two electrons per orbital (Pauli): \(2 \times 16 = 32 = 2n^2\).
- Each subshell has exactly one orbital with \(m_l = 0\), so four orbitals, holding \(4 \times 2 = 8\) electrons.
Units check. These are pure counts. The general results \(n^2\) orbitals and \(2n^2\) electrons per shell are a quick check on any such tally.
Three quick tests
- What are the allowed values of \(l\) for an electron in the \(n=4\) shell?
Answer
\(l = 0, 1, 2, 3\) - How many orbitals are in a d subshell?
Answer
5 (corresponding to \(m_l = -2, -1, 0, +1, +2\)) - Can two electrons in the same atom have the quantum numbers \(n=2, l=1, m_l=0, m_s=+\frac{1}{2}\) and \(n=2, l=1, m_l=0, m_s=+\frac{1}{2}\)?
Answer
No, this violates the Pauli Exclusion Principle because all four quantum numbers are identical.
Back to CU-101 →
Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.