De Broglie Explanation

12.15 The Mystery of Bohr's Second Postulate

Bohr's three postulates work miraculously well for hydrogen. But where does the curious quantisation rule \(L = nh/(2\pi)\) come from? Bohr himself had no answer — it was an inspired guess. Why, of all possible numbers, must angular momentum equal an integer multiple of \(h/2\pi\)?

The answer came in 1924 from Louis de Broglie, while writing his PhD thesis. If electrons have a wave nature (λ = h/p), then a stable orbit is one in which the electron-wave forms a standing wave on the circle. Like a vibrating violin string clamped at both ends, only certain wavelengths are allowed.

12.16 de Broglie's Standing-Wave Argument

Imagine the electron's matter-wave wrapped around a circular orbit of radius \(r_n\). For the wave to interfere constructively with itself after one trip around — that is, for the orbit to be stable — the circumference must contain a whole number of wavelengths:

Substituting de Broglie's wavelength \(\lambda = h/(m_e v)\):

Rearranging:

This is exactly Bohr's quantisation condition! The mysterious "integer angular momentum" is just a fancy way of saying "the matter wave fits the circle smoothly." The mystery is gone; quantum is geometry.

A non-allowed (forbidden) orbit

If the circumference \(2\pi r\) is not a whole number of wavelengths, the wave returns out of phase after each circuit. Successive trips would cancel each other by destructive interference. Over time the wave dies out — the orbit cannot be stable. Only the allowed integer orbits survive, exactly recovering Bohr's quantisation.

12.17 Limitations of Bohr's Model

Bohr's model was a triumph for hydrogen. But its successes ended there. Detailed observations soon revealed cracks:

(a) Limited to one-electron systems

Bohr's formula correctly predicts the energies of hydrogen and hydrogen-like ions (He⁺, Li²⁺, Be³⁺) — any system with one electron orbiting Z protons. For neutral helium (2 electrons), the model fails utterly. Inter-electron repulsion is not included; a complete description requires the many-body Schrödinger equation.

(b) Cannot explain spectral line intensities

Bohr's model says which lines exist, but not how bright each one is. In reality, some Balmer lines are extremely intense, others nearly invisible. Predicting these intensities requires transition probabilities (matrix elements of the dipole moment) — an output of full quantum mechanics, not Bohr's simple postulates.

(c) Fine structure

Examined under high resolution, every "single" Bohr line splits into two or more closely-spaced lines — the so-called fine structure. This splitting (~10⁻⁴ eV) is due to the electron's spin coupling to its orbital motion (spin-orbit interaction) and to relativistic corrections, neither of which appears in Bohr's model.

(d) Zeeman and Stark effects

Place a hydrogen discharge tube in a strong magnetic field: each Bohr line splits into several components — the Zeeman effect. A strong electric field similarly splits lines (Stark effect). Bohr's model has nothing to say about either, since it allows only one possible orbit shape (a circle) at each n.

(e) Semi-classical, not consistently quantum

Bohr postulates that orbits don't radiate (a quantum statement) but otherwise treats the electron as a tiny billiard ball moving on a definite circular path (a classical statement). The two pictures don't sit easily together. The complete quantum-mechanical description, due to Erwin Schrödinger (1926), abandons the idea of definite orbits altogether. Instead, it describes the electron by a wavefunction ψ(r) whose squared modulus |ψ|² gives the probability of finding it at point r. The "orbits" of Bohr are replaced by orbitals — three-dimensional probability clouds.

Worked Examples