Atomic models and Schrödinger
Classical electromagnetism predicts that no atom can last more than sixteen picoseconds. Resolving that contradiction led from Bohr to Schrödinger to Pauli, and ended up explaining why a crystal has energy bands instead of isolated levels.
01The atom that classical physics could not sustain
Rutherford's scattering experiment (1911) —alpha particles against a gold foil, with some bouncing straight back— brought down Thomson's “plum pudding” model and left an atom with almost all of its mass and all of its positive charge concentrated in a tiny nucleus, with the electrons circling around it. The problem is that this atom, calculated with classical electromagnetism, cannot exist.
An accelerating charge radiates energy —it is the same physics that makes an antenna work— and an electron in orbit accelerates all the time, even at constant speed, because it changes direction. Radiating energy, it would lose radius in a spiral until it fell into the nucleus.
02The Bohr atom
In 1913, Bohr solved the problem by decree: he postulated that the electron can only occupy certain orbits, those where its angular momentum is an integer multiple of \( \hbar \), and that in those orbits —and only in those— it does not radiate. It emits light only when jumping from one orbit to another, and that light comes out as a photon with exactly the energy of the difference.
The postulate had —in 1913— no fundamental justification; it was a rule that worked. It was justified by Schrödinger twelve years later, when the quantization of \( L \) came out by itself from requiring that the electron's wave function fit without discontinuities around the nucleus, with no further hypothesis.
You choose the series and the starting level; the level diagram marks the transition and calculates the exact wavelength of the emitted photon.
03The Schrödinger atom: orbitals, not orbits
The Bohr model predicts hydrogen well and fails with any atom of more than one electron. Solving the Schrödinger equation in three dimensions for the hydrogen atom gives something different from an orbit: a probability cloud, \( |\psi|^2 \), which describes where the electron is most likely to be found, without it having a definite trajectory. From that solution come, unasked, four quantum numbers.
| Quantum number | What it fixes | Possible values |
|---|---|---|
| Principal, \( n \) | The energy (approximately, as in Bohr) | 1, 2, 3… |
| Orbital, \( l \) | The shape of the cloud: \( l=0 \) spherical (s), \( l=1 \) with two lobes (p)… | 0 to \( n-1 \) |
| Magnetic, \( m_l \) | The spatial orientation of that shape | \( -l \) to \( l \) |
| Spin, \( m_s \) | An internal degree of freedom of the electron, with no classical analog | \( +\tfrac12 \) or \( -\tfrac12 \) |
Spin does not come out of the Schrödinger equation —Dirac's relativistic version is needed to derive it— but it has to be added by hand for the model to agree with what is observed: each orbital admits exactly two electrons, one of each spin.
04The Pauli exclusion principle
In 1925, Pauli stated the rule that organizes everything above: two electrons in the same atom can never have all four quantum numbers equal. As a consequence, each orbital —each combination of \( n, l, m_l \)— admits at most two electrons, and only if they have opposite spin. That single rule builds the entire periodic table: the electrons of an atom settle into shells and subshells, from the lowest energy upward, until the available electrons are used up.
Without the Pauli principle, all the electrons in an atom —in any atom, in the whole body— would fall to the lowest-energy state, the 1s orbital. Matter as it exists, with volume and with chemistry, depends on that being forbidden: electrons stack upward in energy because they have no other choice. The same idea, carried over to a crystal, is what in the next unit will explain why a metal has free electrons up to a well-defined energy —the Fermi level— and not a single one above it at zero temperature.
05Where it appears in electronics
When many atoms come together to form a crystal, the Pauli principle forces something remarkable: since no electron in the crystal can repeat the state of another, the single atomic level of each isolated atom —say, the 3s of silicon— splits into as many slightly different levels as there are atoms in the crystal. With \( 10^{22} \) atoms per cubic centimeter, those levels end up so close together that they form, for practical purposes, a continuous band of allowed energy.
| Concept from this topic | Where it leads |
|---|---|
| Discrete levels of an isolated atom | Energy bands of a crystal (next two units) |
| Electrons stacked by Pauli up to a certain energy | Fermi level: the boundary between full and empty in a metal or semiconductor |
| Valence shell of the atom (the outermost orbitals) | Valence band; the four valence electrons of silicon build its four covalent bonds |
| Empty orbitals of higher energy | Conduction band: where an electron has to jump in order to conduct |
No concept of a semiconductor —band gap, Fermi level, charge carrier— makes sense without this topic behind it. It is the missing link between “an atom has electrons in shells” and “a crystal has electrons in bands”.
06In the lab
Calculate the first four wavelengths of the Balmer series and compare them with the measured values (Hα 656.3 nm, Hβ 486.1 nm, Hγ 434.0 nm, Hδ 410.2 nm). Identify the color of each one.
Using the rule of filling by increasing energy and the Pauli principle, write the electron configuration of silicon (\( Z=14 \)) and germanium (\( Z=32 \)). Identify how many valence electrons each one has.
Calculate the ionization energy predicted by the Bohr model for singly ionized helium (\( \text{He}^+ \), a single electron, \( Z=2 \)) and compare it with the measured value of 54.4 eV.
07Common mistakes
- Thinking that the electron “orbits” like a planet. The Schrödinger model gives a probability cloud, not a trajectory; Bohr's, although useful, has been superseded on that point since 1926.
- Applying the Bohr model to atoms with more than one electron. It is only exact for one-electron systems: hydrogen and ions such as \( \text{He}^+ \) or \( \text{Li}^{2+} \).
- Confusing the principal quantum number with the number of electrons. \( n \) organizes energy levels; how many electrons there are up to that point is decided by Pauli.
- Forgetting spin when counting the capacity of an orbital. Each orbital holds two electrons, not one.
- Believing that Pauli applies only to electrons in the same atom. The general version —for any set of fermions that share a quantum system, such as the electrons of an entire crystal— is the one that builds the energy bands.
- Thinking that the classical instability of the atom is only a historical curiosity. It is the question that quantum mechanics as a whole had to answer in order to exist as a theory.
08Self-assessment
Why can the classical Rutherford atom not be stable?
Because a charge in orbit accelerates constantly and, according to classical electromagnetism, it should radiate energy and spiral into the nucleus within picoseconds.
What did Bohr postulate to avoid that collapse?
That the electron can only occupy orbits with quantized angular momentum, \( L = n\hbar \), and that in those orbits it does not radiate.
What replaces the orbit in the Schrödinger model?
A probability cloud, the orbital, described by four quantum numbers: \( n, l, m_l, m_s \).
What does the Pauli exclusion principle say?
That two electrons of the same quantum system can never have all four quantum numbers equal, so each orbital admits at most two electrons with opposite spin.
What would happen to matter without the Pauli principle?
All the electrons would fall to the lowest-energy state; there would be no shells, no periodic table, and no chemistry as we know it.
Why do atomic levels become bands inside a crystal?
Because Pauli forbids two electrons in the crystal from sharing the same state: the single level of each isolated atom splits into as many levels as there are atoms, so close together that they form a continuous band.
09Further reading
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). The chapters on the Bohr atom, the Schrödinger solution for hydrogen and the Pauli principle, with the same rigor as the whole unit.
- Niels Bohr. On the Constitution of Atoms and Molecules. Philosophical Magazine, vol. 26, 1913. The original paper in which he postulates the quantization of angular momentum, with no justification other than “it works”.
- Wolfgang Pauli. Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren. Zeitschrift für Physik, vol. 31, 1925. The exclusion principle paper, written before the modern concept of spin existed.
- Charles Kittel. Introduction to Solid State Physics. 8th ed., Wiley, 2004. The opening chapter explicitly connects atomic levels with the bands of a crystal: the exact bridge to the next two units.