The Crisis of Classical Physics and the Origins of Quantum Mechanics

By 1900 classical physics — Newtonian mechanics, Maxwell's electrodynamics and statistical thermodynamics — described almost everything then measurable. The exceptions were small in number but fatal in kind: the spectrum of thermal radiation, the photoelectric effect, the stability and line spectra of atoms, and the low-temperature behaviour of specific heats. Each is a place where a classical calculation gives a definite, unambiguous answer that experiment flatly contradicts. This chapter follows the historical route by which those anomalies forced the introduction of a single new constant, Planck's $h$, and a single new idea, quantisation, and ends with de Broglie's proposal that matter itself is wavelike — the hypothesis the rest of Unit D turns into a theory.

The organising thread is the quantum of action. Wherever a classical continuum is assumed — a continuous range of oscillator energies, a continuous exchange of energy between light and electrons, a continuous set of orbits — the correct description replaces it with a discrete set, spaced by $h$ or $\hbar = h/2\pi$. Classical physics is recovered whenever the relevant actions are large compared with $\hbar$; this is the correspondence principle, and it is the reason the old physics worked so well for so long.

How to read this chapter. Keep three numbers in view: $h = 6.626\times10^{-34}\ \mathrm{J\,s}$, $\hbar = 1.055\times10^{-34}\ \mathrm{J\,s}$, and $hc = 1240\ \mathrm{eV\,nm}$. Every result below is one of these constants tying a particle quantity (energy, momentum) to a wave quantity (frequency, wavelength): $E = h\nu$, $p = h/\lambda$. The "old quantum theory" of Sections 4 and 8 is a scaffold — internally inconsistent, quantitatively right only for the simplest systems — that Chapters 2 and 3 replace with the Schrödinger equation.

1. Blackbody Radiation and the Ultraviolet Catastrophe

A blackbody absorbs all radiation incident on it; in thermal equilibrium at temperature $T$ it re-emits with a spectral energy density $\rho(\nu,T)$ that depends on $\nu$ and $T$ alone, not on the material. Classical physics computes it by counting electromagnetic modes in a cavity — $8\pi\nu^2/c^3$ per unit volume per unit frequency — and assigning each the equipartition energy $k_B T$. The result is the Rayleigh–Jeans law,

which matches the data at low frequency and diverges as $\nu\to\infty$: the total energy $\int_0^\infty \rho\,d\nu$ is infinite. This is the ultraviolet catastrophe — not a small discrepancy but a qualitative absurdity.

Planck's resolution (1900). Planck assumed the cavity walls' oscillators of frequency $\nu$ can exchange energy only in integer multiples of $h\nu$. The average energy of an oscillator is then not $k_B T$ but

obtained from the Boltzmann-weighted sum over $E_n = nh\nu$. Multiplying by the mode density gives the Planck law

For $h\nu\ll k_B T$ the exponential expands to $1 + h\nu/k_B T$ and Planck reduces to Rayleigh–Jeans; for $h\nu\gg k_B T$ the $e^{-h\nu/k_B T}$ factor cuts the spectrum off exponentially, curing the catastrophe. Fitting the curve to data fixed $h$ for the first time.

Worked example — Wien and Stefan–Boltzmann as corollaries. Maximising $\rho(\nu,T)$ gives $\nu_{\max}\propto T$ (Wien's displacement law; in wavelength, $\lambda_{\max}T = 2.898\times10^{-3}\ \mathrm{m\,K}$). Integrating $\rho$ over all $\nu$ gives a total energy density $\propto T^4$, hence a radiated power per unit area $\sigma T^4$ with $\sigma = 2\pi^5 k_B^4/15h^3c^2$ — the Stefan–Boltzmann constant, now expressed through $h$.

2. The Photoelectric Effect

Light striking a clean metal surface ejects electrons. The classical wave picture predicts that the electron energy should grow with intensity and that there should be a measurable delay at low intensity while energy accumulates. Experiment (Lenard, then Millikan) shows the opposite:

• there is a threshold frequency $\nu_0$ below which no electrons are emitted, at any intensity;
• above threshold the maximum kinetic energy is linear in frequency and independent of intensity;
• the emission is instantaneous (within $10^{-9}\ \mathrm{s}$);
• intensity controls only the number of electrons, not their energy.

Einstein's photon hypothesis (1905). Light of frequency $\nu$ consists of quanta each carrying energy $E = h\nu$. One quantum is absorbed by one electron; if $h\nu$ exceeds the work function $\phi$ (the minimum energy to free an electron from the metal), the electron escapes with

The threshold is $\nu_0 = \phi/h$; the slope of $K_{\max}$ versus $\nu$ is $h$, the same $h$ as in the Planck law. Millikan's precision measurement of that slope (1916), undertaken to disprove the idea, instead confirmed it and gave an independent value of $h$.

3. The Compton Effect

When X-rays of wavelength $\lambda$ scatter from (effectively free) electrons, the scattered radiation contains a component at a longer wavelength $\lambda'$ that depends on the scattering angle $\theta$ but not on the target material:

The quantity $h/m_e c = 2.43\ \mathrm{pm}$ is the Compton wavelength of the electron. The derivation treats the photon as a relativistic particle with energy $E = h\nu = pc$ and momentum $p = h/\lambda$, and applies conservation of energy and momentum to a two-body elastic collision. A classical wave, which would drive the electron and be re-radiated at the same frequency, cannot produce a wavelength shift at all. Compton scattering (1923) is the cleanest single demonstration that electromagnetic radiation carries quantised momentum as well as quantised energy.

Worked example — maximum shift and the backscatter peak. The shift is largest at $\theta = 180^\circ$ (photon reversed), where $\lambda'-\lambda = 2h/m_ec = 4.85\ \mathrm{pm}$. For $500\ \mathrm{keV}$ photons this is a large fractional change and the recoil electron takes most of the energy; for visible light ($\lambda\sim 500\ \mathrm{nm}$) the same absolute shift is one part in $10^5$ and utterly negligible — which is why the effect was invisible until X-rays.

4. Atomic Spectra and the Bohr Model

Atoms emit and absorb light only at sharp, discrete wavelengths. For hydrogen the wavenumbers fit the Rydberg formula

with $n_i,n_f$ positive integers. Classically an orbiting electron radiates continuously, spiralling into the nucleus in about $10^{-11}\ \mathrm{s}$: classical atoms are neither stable nor line-emitting.

Bohr's postulates (1913).

1. The electron occupies only certain stationary states, in which — contrary to electrodynamics — it does not radiate.
2. The allowed orbits are those for which the angular momentum is quantised: $L = n\hbar$.
3. Radiation is emitted or absorbed only in transitions between stationary states, with photon energy $h\nu = E_i - E_f$.

Applying $L = n\hbar$ to a circular Coulomb orbit gives the radii $r_n = n^2 a_0$ with the Bohr radius $a_0 = 4\pi\varepsilon_0\hbar^2/m_e e^2 = 0.529\ \text{\AA}$, and the energies

The transition rule $h\nu = E_i - E_f$ then reproduces the Rydberg formula and predicts $R_\infty$ from fundamental constants. The model also fits singly ionised helium (with $Z^2$ scaling) but fails for any two-electron system and says nothing about line intensities or splittings.

5. The de Broglie Hypothesis

If light — historically a wave — has particle properties $E = h\nu$, $p = h/\lambda$, then de Broglie (1924) proposed the symmetric statement: a particle of momentum $p$ has an associated wavelength

and a frequency $\nu = E/h$. For a non-relativistic particle $\lambda = h/\sqrt{2mK}$. This immediately explains Bohr's quantisation condition: a standing wave on the circular orbit requires the circumference to be an integer number of wavelengths, $2\pi r = n\lambda = nh/p$, i.e. $L = pr = n\hbar$. The mysterious postulate becomes a resonance condition.

Worked example — when is the wavelength observable? An electron accelerated through $150\ \mathrm{V}$ has $K = 150\ \mathrm{eV}$ and $\lambda = h/\sqrt{2m_eK} \approx 1.0\ \text{\AA}$ — comparable to atomic spacings in a crystal, so diffraction is observable. A $1\ \mathrm{g}$ bead moving at $1\ \mathrm{cm\,s^{-1}}$ has $\lambda \sim 10^{-29}\ \mathrm{m}$, some $10^{14}$ times smaller than a nucleus: no apparatus can resolve it, and the bead is classical.

6. Matter-Wave Diffraction: the Experimental Confirmation

Davisson and Germer (1927) scattered low-energy electrons from a nickel single crystal and found intensity maxima at exactly the angles predicted by the Bragg condition $d\sin\theta = n\lambda$ using the de Broglie $\lambda$. Independently, G. P. Thomson passed electrons through thin polycrystalline films and photographed Debye–Scherrer rings identical in form to X-ray powder patterns. Later experiments extended matter-wave diffraction to helium atoms, neutrons, and molecules as large as fullerenes and beyond. The single-particle double slit — electrons sent through one at a time — builds an interference pattern dot by dot: each electron arrives as a localised event, yet the accumulated distribution is the wave pattern. Wave and particle aspects are both necessary and never observed in the same measurement.

7. Further Cracks: Specific Heats and the Franck–Hertz Experiment

Specific heat of solids. The classical Dulong–Petit law gives a molar heat capacity $3R$, temperature-independent. Measurements show $C\to 0$ as $T\to 0$. Einstein (1907) modelled the solid as $3N$ oscillators each quantised in units of $h\nu_E$; the same $\langle E\rangle$ formula as in the Planck law gives $C\to 0$ exponentially at low $T$. Debye (1912) improved the low-$T$ behaviour to the observed $C\propto T^3$ by quantising a spectrum of lattice modes. Either way, quantisation is what freezes out degrees of freedom.

Franck–Hertz (1914). Electrons accelerated through mercury vapour lose energy in fixed lumps of $4.9\ \mathrm{eV}$, and the vapour then emits ultraviolet light at exactly the corresponding wavelength. This is a direct, non-spectroscopic demonstration that atomic internal energy is quantised: the collision is inelastic only when the electron can deliver the precise excitation energy.

8. What the Old Quantum Theory Could and Could Not Do

The Bohr–Sommerfeld programme — quantise the action $\oint p\,dq = nh$ for each periodic coordinate — extended Bohr's model to elliptical orbits and gave the fine structure of hydrogen to leading order. But it is a patchwork:

• it applies only to systems whose classical motion is multiply periodic, i.e. essentially one-body problems with special symmetry;
• it cannot treat the ground state of helium, the hydrogen molecule ion, or any non-separable system;
• it gives energies but not transition rates or selection rules, which have to be grafted on by a correspondence argument;
• it retains classical trajectories, which the uncertainty principle (Chapter 5) will show cannot exist.

What survives into the real theory is the physical core: energy levels are discrete, transitions are quantum jumps with $h\nu = \Delta E$, angular momentum comes in units of $\hbar$, and every particle has a wave associated with it. The task of Chapters 2–4 is to find the equation that wave obeys and the rules for extracting predictions from it.

9. Common Pitfalls

• Calling the photoelectric work function an ionisation energy. $\phi$ is the energy to remove an electron from a metal surface to just outside it; it is a bulk property, typically $2$–$5\ \mathrm{eV}$, not the ionisation energy of an isolated atom.
• Thinking higher intensity means higher photon energy. Intensity is photons per second times $h\nu$; the energy per photon is set by frequency alone. A bright red lamp will never eject electrons that a dim blue one can.
• Using the non-relativistic $\lambda = h/\sqrt{2mK}$ for fast electrons. For fast electrons use the relativistic $p = \sqrt{E^2/c^2 - m^2c^2}$ in $\lambda = h/p$; the non-relativistic $\lambda = h/\sqrt{2mK}$ is only a low-energy approximation.
• Treating the Compton shift as fractional. $\Delta\lambda = (h/m_ec)(1-\cos\theta)$ is an absolute shift, the same in picometres for X-rays and for visible light; it is only detectable when it is a decent fraction of $\lambda$.
• Believing the Bohr model "explains" the atom. It gets hydrogen's energy levels right by a lucky cancellation and fails for everything else; its orbits are not real. Use it as mnemonic, not mechanism.
• Saying an electron is "sometimes a wave and sometimes a particle." In any single experiment it shows one aspect; the wavefunction (next chapter) is a single object that carries both, with $|\psi|^2$ giving the probability of a particle-like detection.
• Forgetting the correspondence limit. Quantum results must reduce to classical ones for large quantum numbers / large actions; if a formula does not, it is wrong.

Summary

• Planck: cavity oscillators have $E_n = nh\nu$, giving $\rho(\nu,T) = \dfrac{8\pi\nu^2}{c^3}\dfrac{h\nu}{e^{h\nu/k_BT}-1}$; reduces to Rayleigh–Jeans for $h\nu\ll k_BT$, cuts off exponentially for $h\nu\gg k_BT$. Yields Wien ($\lambda_{\max}T$ const) and Stefan–Boltzmann ($\sigma T^4$).
• Photoelectric effect: $K_{\max} = h\nu - \phi$; threshold $\nu_0 = \phi/h$; energy independent of intensity, emission instantaneous — light delivers energy in quanta $h\nu$.
• Compton effect: $\lambda' - \lambda = \dfrac{h}{m_ec}(1-\cos\theta)$, from photon–electron relativistic two-body kinematics with $p_{\text{photon}} = h/\lambda$.
• Bohr model: $L = n\hbar \Rightarrow r_n = n^2a_0$, $E_n = -13.6\ \mathrm{eV}/n^2$; $h\nu = E_i - E_f$ reproduces the Rydberg formula and predicts $R_\infty$.
• de Broglie: $\lambda = h/p$; standing-wave condition on a Bohr orbit gives $L = n\hbar$. Confirmed by Davisson–Germer and G. P. Thomson electron diffraction, and by single-particle double-slit build-up.
• Other evidence: specific heats $\to 0$ at low $T$ (Einstein/Debye); Franck–Hertz shows discrete atomic excitation energies.
• Old quantum theory ($\oint p\,dq = nh$) works only for multiply periodic one-body systems and gives no transition rates; the surviving physical core is discreteness, quantum jumps, $\hbar$-quantised angular momentum, and universal wave–particle duality.