Why we need relativistic quantum mechanics

At the end of the last chapter we said: the entire edifice of quantum mechanics stands on the non-relativistic foundation $E=p^2/2m$. This section starts with a practical question — when does that foundation give way? The answer is much closer to home than most people expect: no collider required, the gold ring on your hand contains electrons moving at nearly the speed of light.

Opening with a phenomenon: why is gold golden?

In the periodic table, silver is white, copper is red, gold is yellow. Silver and gold sit in the same group with the same electron configuration pattern ($d^{10}s^1$) — so why do their colours differ so much?

A metal’s colour is set by “which light the electrons absorb”. Gold’s yellow comes from the $5d\to6s$ transition: that gap happens to fall at the blue-violet end of the visible spectrum (about 2.4 eV), so blue light is absorbed and the reflected light leans yellow. Silver’s corresponding gap sits in the ultraviolet (about 3.7 eV), all visible light is reflected, and so it gleams white.

The trouble is: a non-relativistic Schrödinger calculation puts gold’s gap in the ultraviolet too — computed gold comes out silvery white. Where is the discrepancy?

It lies in gold’s nucleus of 79 protons. Estimate the speed of the innermost (1s) electron with the Bohr model (for a hydrogen-like 1s orbit around $Z$ protons, $v=Z\alpha c$, with $\alpha\approx1/137$ the fine-structure constant):

$$\frac{v}{c}\approx Z\alpha=\frac{79}{137}\approx0.58\tag{12.1.1}$$

Sixty percent of the speed of light. At that speed, relativistic effects are no longer corrections in the decimal places: the mass-increase factor is $\gamma=1/\sqrt{1-0.58^2}\approx1.22$, i.e. the “moving mass” is 22% heavier. A heavier electron means an orbital radius $\propto1/m$ that contracts, and the $s$ orbitals (largest probability near the nucleus, feeling the effect most strongly) contract hardest; $6s$ is pulled down, $5d$ is pushed up by the altered inner screening — the gap shrinks and drops from the ultraviolet right into the blue end of the visible. The colour of gold is a relativistic effect visible to the naked eye.

The same mechanism explains mercury: its $6s^2$ closed shell is relativistically contracted so low and tight that it barely shares electrons with its neighbours, the interatomic bonding is anomalously weak — and so mercury is the only metal that is liquid at room temperature. Without relativity, mercury would be a solid at room temperature.

The formal level: how the Schrödinger equation offends relativity

Phenomena done, now the structure. The heart of special relativity is that time and space stand on equal footing: Lorentz transformations mix $t$ and $x$, so a covariant equation must treat them alike. Now lay out the Schrödinger equation:

$$\ii\hbar\,\frac{\partial\Psi}{\partial t} =-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi\tag{12.1.2}$$

Estimate: when the corrections stop being negligiblebasic~6 minexpand

Take the ratio of the correction to the leading term as the criterion:

$$\frac{p^4/8m^3c^2}{p^2/2m}=\frac{p^2}{4m^2c^2}=\frac14\Bigl(\frac vc\Bigr)^2\tag{12.1.5}$$

Inner electron of a hydrogen-like atom. In the first Bohr orbit, Coulomb attraction supplies the centripetal force: $\frac{Ze^2}{4\pi\epsilon_0 r^2}=\frac{mv^2}{r}$, and combined with angular-momentum quantisation $mvr=\hbar$, dividing the two equations gives

$$v=\frac{Ze^2}{4\pi\epsilon_0\hbar}=Z\alpha c,\qquad \alpha\equiv\frac{e^2}{4\pi\epsilon_0\hbar c}\approx\frac1{137}\tag{12.1.6}$$

• Hydrogen: $(v/c)^2\approx5\times10^{-5}$ → corrections of order $10^{-5}$, which times the Rydberg energy 13.6 eV is $10^{-4}$ eV — precisely the measured size of the fine structure. Consistent.
• Gold: $(v/c)^2\approx0.33$ → the “correction” reaches 8%, with higher orders crowding right behind — patching order by order loses its meaning, and a relativistic equation is needed in one stroke.

Another road to the same conclusion. To localise an electron within a scale $\Delta x$, the uncertainty principle demands $\Delta p\gtrsim\hbar/\Delta x$, giving an energy fluctuation $\sim\Delta p\,c\gtrsim\hbar c/\Delta x$. Once $\Delta x$ shrinks to

$$\lambda_C\equiv\frac{\hbar}{mc}\approx3.9\times10^{-13}\ \text{m (electron)}\tag{12.1.7}$$

(called the Compton wavelength), the fluctuation energy reaches the order of $mc^2$ — enough to conjure particle-antiparticle pairs out of nothing. On scales below the Compton wavelength, the very concept of “the wavefunction of one particle” begins to fail.

∎Key formulas

• Relativistic dispersion relation

  $$E^2=p^2c^2+m^2c^4$$

  Low-speed expansion: mc² + p²/2m − p⁴/8m³c² + …; Schrödinger kept only the second term

• Inner-electron speed

  $$\frac{v}{c}\approx Z\alpha,\qquad \alpha\approx\frac1{137}$$

  Hydrogen 0.7%; gold 58% — the relativistic dividing line in chemistry

• Size of relativistic corrections

  $$\frac{\Delta E}{E}\sim\Bigl(\frac vc\Bigr)^2\sim(Z\alpha)^2$$

  This is where hydrogen’s ∼10⁻⁴ eV fine structure comes from

• Compton wavelength

  $$\lambda_C=\frac{\hbar}{mc}\approx3.9\times10^{-13}\ \text{m}$$

  Localise below this scale → energy fluctuations ≳ 2mc² → the single-particle picture fails

?Self-check4 questions

1. 1.

   The direct structural conflict between the Schrödinger equation and special relativity is:

   • The time derivative is first order while the space derivative is second order, so the form is not preserved under Lorentz transformations
   • The imaginary unit i appears in the equation
   • The equation contains a potential term V(x)
   • The wavefunction is complex while relativity requires real fields

   Submit
2. 2.

   Gold being golden and mercury being liquid share which microscopic origin?

   • Heavy nuclei drive inner electrons to Zα·c ≈ 0.6c; the relativistic mass increase contracts the s orbitals markedly and shifts the levels
   • The nuclei of gold and mercury undergo weak radioactive decay
   • Electrons in heavy atoms have larger spin
   • Coulomb repulsion between electrons is stronger in heavy atoms

   Submit
3. 3.

   Why does the high-energy regime shake not merely the Schrödinger equation but the "single-particle wavefunction" framework itself? (Select all that apply.)

   Select all that apply

   • Above 2mc² particle-antiparticle pairs can be created, so the particle number is no longer fixed
   • The normalisation ∫|Ψ|²=1 and probability conservation assume, at their core, that the particle number is always 1
   • Localising a particle below its Compton wavelength gives energy fluctuations large enough to create new particles
   • At high energy the position operator ceases to be Hermitian

   Submit
4. 4.

   Use v/c ≈ Zα to estimate the ratio of the 1s electron speed to the speed of light in gold (Z=79). (α=1/137.0)

   2% relative tolerance

   Submit

What comes next

Diagnosis complete — time to write the prescription. The most direct idea: since the disease is $E=p^2/2m$, swap in the genuine article $E^2=p^2c^2+m^2c^4$ and build a new equation by the same translation rules. That equation is the Klein-Gordon equation — it cures the covariance, and instantly detonates something stranger: probabilities can go negative.