Quantum Numbers

Solving the Schrödinger equation for an atom does not give a single wave function but a whole family of them, each one labelled by a specific combination of quantum numbers. Four quantum numbers, taken together, completely specify the state of an electron in an atom (their allowed values and physical meaning are summarised in the table below).

Principal quantum number ($n$). Introduced already by Bohr, $n$ can take any positive integer value $1, 2, 3, \ldots$ and identifies the shell an orbital belongs to. It is the chief factor controlling both the average distance of the electron from the nucleus (larger $n$ means, on average, farther from the nucleus) and, for a given atom, the orbital's energy: for a one-electron species, energy depends on $n$ alone, while for a many-electron atom it depends on both $n$ and $l$.

Azimuthal (orbital angular momentum) quantum number ($l$). For a given $n$, $l$ can take any integer value from $0$ to $(n-1)$, giving $n$ possible values in total. $l$ identifies the subshell (Section 2.3) and, most importantly, fixes the shape of the orbital's boundary surface: $l = 0$ gives a spherical (s) orbital, $l = 1$ gives a dumbbell-shaped (p) orbital, $l = 2$ gives a (mostly) four-lobed (d) orbital, and $l = 3$ gives a still more complex (f) orbital.

Magnetic quantum number ($m_l$). For a given $l$, $m_l$ can take any integer value from $-l$ through $0$ up to $+l$, giving $(2l+1)$ possible values -- which is exactly the number of orbitals in that subshell (1 for s, 3 for p, 5 for d, 7 for f). Physically, $m_l$ fixes the spatial orientation of an orbital: for instance, the three values $m_l = -1, 0, +1$ of the $p$ subshell correspond to the three $p$ orbitals ($p_x$, $p_y$, $p_z$) pointing along the three different Cartesian axes.

Spin quantum number ($m_s$). Unlike the first three, spin does not come from solving the spatial Schrödinger equation; it is an intrinsic property of the electron itself, rather like a tiny electron spinning on its own axis (though this is only a classical analogy, not a literal description). An electron can have only one of two spin states, conventionally written $m_s = +\tfrac{1}{2}$ (spin "up", $\uparrow$) or $m_s = -\tfrac{1}{2}$ (spin "down", $\downarrow$). …