Discovery of the Relativistic Schrödinger Equation

This paper discusses Erwin Schrödinger's discovery of the relativistic wave equation for a spin-zero charged particle in a Coulomb field and explains the reasons behind his decision not to publish it.

The Gist

Here is an explanation of the paper "Discovery of the Relativistic Schrödinger Equation," translated into everyday language with some creative analogies.

The Big Picture: A "What If" Story

Imagine you are a master chef (Erwin Schrödinger) trying to invent a new recipe for a perfect soufflé (the atom). You know the basic ingredients (electrons and protons), and you have a new theory about how they dance (de Broglie's matter waves).

In late 1925, a colleague (Peter Debye) told Schrödinger, "Hey, if these things are waves, you need a recipe book (an equation) to describe how they move." Schrödinger said, "Challenge accepted!"

He went to work and wrote down a "Super-Recipe" that included the rules of Einstein's Relativity (things moving very fast). He thought, "This is it! This explains everything!" But when he cooked the dish and tasted it, it was disgusting. The flavor didn't match reality. The soufflé collapsed.

This paper is a historical detective story. The authors are digging through Schrödinger's old notebooks and letters to answer a burning question: Why did Schrödinger throw away his "Relativistic Recipe" and only publish the "Non-Relativistic Recipe" that made him famous?

The Plot Twist: The Missing Ingredient

The paper reveals that Schrödinger did find the equation for a fast-moving electron, but it gave him the wrong answer for the hydrogen atom.

The Analogy of the Blindfolded Archer:
Imagine Schrödinger was an archer trying to hit a target (the atom's energy levels).

• The Target: The actual energy levels of a hydrogen atom, which scientists had measured very precisely.
• The Arrow: Schrödinger's new Relativistic Equation.
• The Problem: The arrow missed the target by a wide margin. It was off by a factor of 8/3!

Why did it miss?
The authors explain that Schrödinger was aiming at a target that had a secret feature he couldn't see: Electron Spin.

• Think of the electron not just as a ball, but as a spinning top.
• In 1925, nobody knew about this "spin." Schrödinger treated the electron like a simple, non-spinning marble.
• Because he ignored the spin, his "Relativistic Recipe" produced a result that was mathematically beautiful but physically wrong.

The "Christmas Vacation" Detour

The paper details a famous story: Schrödinger went on a Christmas vacation to a ski resort in Arosa, Switzerland. He took his "failed" relativistic equation with him.

He realized, "This equation is too complicated, and the answer is wrong." So, he made a bold decision. He stripped away the "Relativity" part (the Einstein rules) and focused only on the slow-moving, non-relativistic version.

The Result:
He solved the simpler version. It matched the experimental data perfectly. He published this as the famous Schrödinger Equation (the one every physics student learns today).

He essentially said, "I'll publish the part that works, and I'll hide the part that failed until I figure out what's missing."

The "Ghost" Equation

The paper highlights that the equation Schrödinger found but didn't publish is actually a real, valid equation. It's called the Klein-Gordon equation (named after others who rediscovered it later).

• The Analogy: Imagine Schrödinger found a map to a treasure chest, but the map led to a swamp. He threw the map away. Years later, other explorers found the same map, realized the swamp was just a specific type of terrain, and figured out how to navigate it.
• The authors show that if you solve Schrödinger's "failed" equation correctly, you get a specific set of energy levels. However, because the electron does spin, nature follows a different set of rules (the Dirac Equation, discovered later by Paul Dirac).

The "Spin" Revelation

The paper explains that the reason Schrödinger's relativistic attempt failed was that he didn't know about Spin.

• Schrödinger's View: The electron is a point particle.
• Reality: The electron is a spinning top.
• The Consequence: When you add spin to the math, the "wrong" answer becomes the "right" answer (the Dirac equation). Without spin, the math predicts a fine structure (tiny splits in energy levels) that is 8/3 times too big compared to what we see in real life.

The "Letter to Weyl" Drama

The paper includes a fascinating translation of a letter Schrödinger wrote in 1931 to his friend, the mathematician Hermann Weyl.

In this letter, Schrödinger gets a bit grumpy. He is annoyed that Weyl's book credits Louis de Broglie with discovering the wave equation for particles in a field. Schrödinger argues, "Hey, I was the one who actually wrote down the equation with the electric field in it! De Broglie just talked about waves in empty space."

It's like a chef saying, "You gave credit to the guy who invented the concept of 'soup,' but I'm the one who actually figured out how to cook it with salt and pepper!" Schrödinger was worried about who got the credit for the "Relativistic Wave Equation," even though he had abandoned it years earlier because it didn't work for electrons.

The Conclusion: Why This Matters

The authors conclude that Schrödinger was a genius, but he was also a pragmatist.

1. He found the "Relativistic Equation" first.
2. He realized it was wrong because it didn't account for electron spin (which wasn't discovered yet).
3. He wisely decided to publish the "Non-Relativistic Equation" instead, which worked perfectly for the hydrogen atom and launched the quantum revolution.
4. He kept the "Relativistic Equation" in his back pocket, knowing it was mathematically sound, even if it didn't describe the electron correctly at that time.

The Takeaway:
Science isn't just about being right the first time. Sometimes, it's about realizing you are wrong, understanding why you are wrong (missing the "spin"), and having the courage to pivot to a solution that works. Schrödinger's "failure" with the relativistic equation was actually a crucial step that helped clear the path for the success of quantum mechanics.

As the paper quotes Heinrich Hertz: "Mathematical formulae have an independent existence and an intelligence of their own... we get more out of them than was originally put into them." Schrödinger put a lot of effort into a formula, and even though he thought it failed, the formula itself taught the world a valuable lesson about the nature of the universe.

Technical Summary

Here is a detailed technical summary of the paper "Discovery of the Relativistic Schrödinger Equation" by Barley, Vega-Guzmán, Ruffing, and Suslov.

1. Problem Statement

The paper addresses a historical and mathematical gap in the narrative of quantum mechanics: the specific derivation and fate of the relativistic wave equation for a spin-zero charged particle (specifically in a Coulomb field) by Erwin Schrödinger in late 1925.

While the standard historical narrative focuses on Schrödinger's discovery of the non-relativistic wave equation and the subsequent "failure" of his relativistic attempt due to the lack of electron spin, this paper seeks to:

• Reconstruct the mathematical derivation of the relativistic equation Schrödinger likely derived.
• Solve the equation explicitly for the hydrogen atom (Coulomb potential) to obtain the exact energy levels and eigenfunctions.
• Analyze why this specific relativistic formulation yielded incorrect fine-structure results compared to experimental data (Sommerfeld's formula).
• Clarify the mathematical reasons Schrödinger abandoned this approach in favor of the non-relativistic approximation, prior to the introduction of electron spin by Uhlenbeck and Goudsmit.

2. Methodology

The authors employ a rigorous mathematical reconstruction of Schrödinger's potential path, utilizing modern analytical techniques that were not fully systematized at the time but are applicable to the differential equations Schrödinger faced.

• Derivation of the Equation: Starting from the relativistic energy-momentum relation $(E-e\phi)^2 = (c\mathbf{p}-e\mathbf{A})^2 + m^2c^4$, the authors apply the standard operator substitutions ($E \to i\hbar\partial_t$, $\mathbf{p} \to -i\hbar\nabla$) to derive the second-order Klein-Fock-Gordon type equation for a spin-zero particle.
• Separation of Variables: The equation is separated into radial and angular parts in spherical coordinates. The angular part yields spherical harmonics, while the radial part is reduced to a differential equation for the function $u(x)$.
• Nikiforov-Uvarov (NU) Method: The core of the technical analysis uses the Nikiforov-Uvarov method, a systematic technique for solving second-order linear differential equations of hypergeometric type. This allows for the exact determination of:
  • The quantization condition (energy eigenvalues).
  • The eigenfunctions (expressed in terms of Laguerre polynomials).
  • The normalization constants.
• Semiclassical (WKB) Approximation: The authors also apply the WKB method (specifically using Langer's transformation to handle the singularity at the origin) to the relativistic radial equation. This demonstrates how the exact quantum mechanical result can be recovered from the old Bohr-Sommerfeld quantization rules, bridging the gap between "old" and "new" quantum mechanics.
• Non-Relativistic Limit: The authors perform a Taylor expansion of the relativistic energy formula in the limit $c \to \infty$ (or $\mu \to 0$) to compare the fine-structure corrections with both the non-relativistic Schrödinger theory and the Dirac theory (which includes spin).

3. Key Contributions and Results

A. Exact Solution of the Relativistic Spin-Zero Equation

The paper provides the explicit solution for the relativistic Schrödinger equation for a spin-zero particle in a Coulomb field.

• Energy Levels: The exact energy eigenvalues are derived as:
  $$E_{n_r} = mc^2 \left[ 1 + \left( \frac{\mu}{n_r + \nu + 1} \right)^2 \right]^{-1/2}$$
  where $\mu = Ze^2/\hbar c$ and $\nu = -1/2 + \sqrt{(l+1/2)^2 - \mu^2}$.
• Eigenfunctions: The radial wavefunctions are shown to be proportional to Laguerre polynomials $L_n^{2\nu+1}$, normalized explicitly.

B. The "Fine Structure" Discrepancy

The paper mathematically demonstrates why Schrödinger's relativistic equation failed to match experimental data for hydrogen-like atoms:

• Expansion: When expanded for large $c$, the energy levels yield a fine-structure term:
  $$E \approx mc^2 - \frac{mc^2 \mu^2}{2n^2} - \frac{mc^2 \mu^4}{2n^4} \left( \frac{n}{l+1/2} - \frac{3}{4} \right) + O(\mu^6)$$
• Comparison:
  • Schrödinger (Spin-0): The fine-structure splitting depends on the orbital angular momentum $l$.
  • Sommerfeld/Dirac (Spin-1/2): The correct fine-structure (matching experiment) depends on the total angular momentum $j$.
• Quantitative Error: For the $n=2$ level, the total splitting predicted by the spin-zero relativistic equation is 8/3 times larger than the experimental value (and Sommerfeld's formula). This significant deviation is identified as the primary reason Schrödinger discarded the relativistic equation.

C. Historical Reconstruction

The authors argue that Schrödinger likely:

1. Derived the relativistic equation and the associated energy formula (possibly using the Bohr-Sommerfeld rules or complex integration methods available at the time).
2. Observed the discrepancy with experimental fine-structure data (specifically for ionized helium).
3. Realized that the missing ingredient was electron spin (which was not yet part of quantum mechanics in late 1925).
4. Consequently, he abandoned the relativistic attempt to focus on the non-relativistic limit, which successfully reproduced the gross structure of the hydrogen spectrum without the incorrect fine-structure term.

D. Clarification of Attribution

The paper analyzes a 1931 letter from Schrödinger to Hermann Weyl, clarifying Schrödinger's stance on the attribution of the relativistic scalar equation. Schrödinger objected to the equation being labeled "de Broglie's" in Weyl's book, asserting that while de Broglie proposed matter waves, the specific inclusion of potentials in the wave equation and the scalar relativistic form were his own contributions.

4. Significance

• Mathematical Completeness: The paper fills a gap in the literature by providing the complete, modern mathematical solution to the relativistic equation Schrödinger derived but never published, confirming that the solution was mathematically tractable but physically incorrect for electrons due to the missing spin.
• Historical Insight: It offers a compelling explanation for Schrödinger's "false start." It suggests his decision to publish the non-relativistic equation was a pragmatic scientific choice driven by the immediate mismatch with experimental fine-structure data, rather than a lack of mathematical ability.
• Pedagogical Value: By applying the Nikiforov-Uvarov method and WKB approximation to this historical problem, the paper serves as a bridge between the "old" quantum mechanics (Bohr-Sommerfeld) and modern quantum mechanics, showing how the quantization rules of the former emerge as semiclassical limits of the latter.
• Contextualizing the Spin Discovery: The work highlights the critical timing of the discovery of electron spin (Uhlenbeck and Goudsmit, Nov 1925). Had Schrödinger known about spin, he might have successfully derived the correct relativistic equation (the Dirac equation) earlier, or at least understood why his spin-zero model failed.

In summary, the paper reconstructs the "lost" relativistic chapter of Schrödinger's discovery, proving that while the equation was mathematically sound for a spin-zero particle, it was physically inadequate for the electron, necessitating the shift to the non-relativistic approximation that launched the wave mechanics revolution.