Bohr's neat orbits could not survive two revolutionary ideas of the 1920s: that matter has a wave nature, and that the very act of locating a tiny particle disturbs it. Together these led to the quantum mechanical model, in which an electron is not a planet on a track but a smeared-out probability cloud described by a wavefunction.
In 1924 Louis de Broglie proposed that every moving particle has an associated wavelength $\lambda=\frac{h}{mv}=\frac{h}{p}$, where $h=6.626\times10^{-34}\,\text{J s}$ is Planck's constant and $p=mv$ is momentum. For everyday objects the mass is so large that $\lambda$ is utterly negligible, which is why we never see a cricket ball diffract. For an electron, however, $\lambda$ is comparable to atomic spacings — and electron diffraction (Davisson and Germer) confirmed it. De Broglie's relation also explains Bohr's quantisation: a stable orbit must contain a whole number of electron wavelengths, $2\pi r=n\lambda$, which rearranges to $mvr=\frac{nh}{2\pi}$.
In 1927 Werner Heisenberg showed there is a fundamental limit on how precisely we can simultaneously know a particle's position and momentum: $\Delta x\cdot\Delta p\ge\frac{h}{4\pi}$. The more sharply we pin down position ($\Delta x$ small), the more uncertain the momentum ($\Delta p$ large), and vice versa. This is not a measuring-instrument flaw but a property of nature. It demolishes the idea of a definite electron orbit — if position were exactly known, momentum would be completely undefined. So instead of a path we speak of the probability of finding the electron in a region, described by an orbital: a three-dimensional region around the nucleus where the probability of finding an electron is high (conventionally $90\%$).
Erwin Schrödinger's wave equation, solved for the hydrogen atom, gives wavefunctions $\psi$ labelled by three quantum numbers; a fourth describes the electron's own spin:
- Principal quantum number $n$ ($=1,2,3,\dots$): fixes the main energy level / shell and the size of the orbital. The shell holds at most $2n^2$ electrons.
- Azimuthal (angular momentum) quantum number $l$ ($=0$ to $n-1$): fixes the sub-shell and the shape of the orbital. $l=0,1,2,3$ are the $s,p,d,f$ sub-shells.
- Magnetic quantum number $m_l$ ($=-l$ to $+l$, including $0$): fixes the orientation of the orbital in space, giving $2l+1$ orbitals per sub-shell.
- Spin quantum number $m_s$ ($=+\tfrac{1}{2}$ or $-\tfrac{1}{2}$): the two possible spin states of an electron.
The shapes follow from $l$. An $s$ orbital ($l=0$) is spherically symmetric. A $p$ orbital ($l=1$) is dumb-bell shaped with two lobes along an axis, and there are three of them ($p_x,p_y,p_z$). The five $d$ orbitals ($l=2$) have more complex four-lobe (and one doughnut-plus-lobes) shapes. Each orbital has a node — a surface where the probability of finding the electron is zero. There are two kinds: radial nodes (spherical, counted by $n-l-1$) and angular nodes (planar/conical, counted by $l$). The total number of nodes is $n-1$. For example a $2s$ orbital has $2-0-1=1$ radial node and $0$ angular nodes; a $2p$ orbital has $0$ radial and $1$ angular node (the nodal plane through the nucleus separating its two lobes).
This probabilistic, wave-based picture replaces fixed Bohr orbits while keeping the quantised energy levels that explained the hydrogen spectrum — but now it works for many-electron atoms too, because the quantum numbers give a complete, consistent address for every electron.