*In this installment on the history of atom theory, physics professor (and my dad) Dean Zollman features Erwin Schroedinger—best known for his thought experiment involving a box and a cat. Schroedinger found a way to visualize quantum ideas. Heisenberg, who developed a more complex approach, was not pleased.—Kim*

**By Dean Zollman**

As I discussed last time, Heisenberg developed his ideas without any direct reference to wave-particle duality that had been postulated by Louis de Broglie. A different approach was taken by Erwin Schroedinger (1887-1961). He took the wave ideas to heart and began working on a theory in which an electron in the hydrogen atom behaved somewhat as a vibrating string. As shown in the diagram below, a string which is fixed at both ends can support only certain vibrations which are related to the length of the string. Thus this phenomena seems to have some relation to the Bohr atom with its limit on the number energies.

Three different modes of vibrations for a string which is held fixed at both ends. (By Christophe Dang Ngoc, CC-BY-SA-3.0 via Wikimedia Commons)

Schroedinger needed to combine this idea with the de Broglie hypothesis to obtain a mathematical formulation for atoms and other small objects. As with Heisenberg, Schroedinger’s breakthrough would come when he took time away from his daily grind. However, his motivation was quite different from Heisenberg’s hay fever. I will rely on Arthur I. Miller, a historian of science, to describe it.

“A good friend of Erwin Schroedinger recalled that ‘he did his greatest work during a late erotic outburst in his life.’ The epiphany occurred in the Christmas holidays of 1925 when the thirty-eight-year-old Viennese physicist vacationed with his former girlfriend at the Swiss ski resort of Arosa near Davos. Their passion was the catalyst for a year-long creative activity.”

(Most historians suspect that Schroedinger’s wife, Annemarie (1896-1965), would have been aware of this liaison. Schroedinger was a well-known womanizer.)

Erwin Schroedinger in 1933 (Nobel Foundation, Public domain via Wikimedia Commons)

Schroedinger constructed his equation by using de Broglie’s concept and analogies with optics and other wave phenomena. The result was a differential equation in which one can enter information about the energy of the particle. Then solving the equation yields a wave function which provides some information about the particle’s motion. I am being deliberately vague because at the time it was not clear to Schroedinger or his colleagues exactly what information the wave function was providing.

Prior to the creation of the euro, Schroedinger appeared on the Austrian 1000 shilling note. (By Oesterreichische Nationalbank, Public domain, via Wikimedia Commons)

**Visualization**

Schroedinger’s equation was more appealing to physicists than Heisenberg’s matrix formulation. First, differential equations, while they can be difficult to solve, were well known entities. Newton’s Second Law is an example of a differential equation which physicists had been dealing with for about 200 years. Second, it was much easier to use in calculations than the matrix approach.

As I mentioned last time, using matrices Wolfgang Pauli needed 40 pages of calculations to obtain numbers for the energy levels in the hydrogen. With the Schroedinger approach, a couple of pages is sufficient. The solution, the wave function, can be visualized. For example, the sketch below shows part of the wave function solution for a beam of electrons striking a very thin metal plate. The top drawing represents the electron energy (blue line) with the thin metal represented by the black line. The bottom drawing is the wave function when the information from the top drawing is put into Schroedinger’s equation. (These drawings are from one of my teaching projects, *Visual Quantum Mechanics*.)

Part of a wave function. (From the Quantum Tunneling program of Visual Quantum Mechanics.)

In the early days, the wave function was thought to represent the location of the charge on the electron or the distribution of the electric charge in space. Neither were very satisfying. Eventually Max Born suggested that the square of the wave function represents the probably of finding the electron at each point in space. That interpretation of the wave function did not have a strong theoretical foundation but it stuck and made calculations using Schroedinger’s equation very valuable and useful in a variety of areas of physics and chemistry.

In describing both Heisenberg’s and Schroedinger’s approach I have used words such as *developed* or *constructed*; I have avoided *derived*. In physics and mathematics, we generally think about fundamental laws being derived. We start with some principles that are well established, bring them together, maybe make a few assumptions, and derive some new ideas.

For both the matrix and wave approach to quantum physics, this was not the case. To get to useful results, Schroedinger, Heisenberg, and their colleagues used a variety of analogies and other operations that made sense but could not be derived. Their work is the basis for essentially all of the physics and chemistry related to very small objects. Yet, it cannot be derived; it just works.

**Rivalry**

Within a few years it was clearly shown that the two approaches were equivalent and led to the same conclusions. However, that did not make Heisenberg and Schroedinger friends. Publicly and privately, they criticized each other. A statement from Schroedinger about the origin of his work says,

My theory was inspired by L. de Broglie … and by short but incomplete remarks by A. Einstein. … No genetic relationship whatever with Heisenberg is known to me. I knew of his theory, of course, but felt discouraged, not to say repelled, by the methods of transcendental algebra which appeared very difficult to me and by the lack of visualizability.

Schroedinger was repelled by matrix mathematics (which he called “transcendental algebra”) and the lack of a visual connection. Heisenberg, in a letter to Pauli, was somewhat stronger in his views and Schroedinger’s reliance on visualization.

The more I reflect on the physical portion of Schroedinger’s theory the more disgusting I find it. What Schrodinger writes on the visualizability of his theory is probably not quite right. In other words, it’s crap.

Of course, the letter was written in German. The translation here comes from a chapter by Arthur I. Miller. I have seen the last word (*crap*) translated in a variety of ways*—*from *poppycock* to *bullsh***. The German word was *Mist* which is generally translated as *manure*. In today’s usage, at least among my German friends, *crap* is a good translation. Some of the others are too mild, and some are too strong.

**Both Approaches Have Their Place**

While feelings ran high in the 1920s, both approaches are now considered very valuable. Physicists choose which to use based on what type of problem they need to solve. For most teaching situations, the wave function approach is introduced first because of its visualization capabilities. However, in some recent advanced undergraduate courses instructors have been starting with part of the matrix method.

Most importantly, quantum physics was a revolution in the way we think about matter. It provides the foundation for our understanding and allows engineers and scientists to develop and design many of our modern devices. What started with the ancient Greeks’ attempts to understand matter reached a milestone thousands of years later with the development of quantum physics.

There are still some fundamental unanswered questions about quantum physics. Nobel Laureate Richard Feynman (1918-1988) famously said, “I think I can safely say that nobody understands *quantum mechanics*.” But it has worked well for almost 100 years to explain many phenomena related to atoms, molecules, and solids.

When I began this series almost four years ago, I started with ideas from a short lecture that I had given at the Smithsonian Institution in the 1990s. Based on that talk, I expected to write about six to eight posts and be done. But I found a lot of interesting distractions along the way. Now that I have finally reached the quantum revolution, I will take a break. A lot of interesting developments have occurred in the past 90 years. Before I think more about them, I will pause for a while.

**Dean Zollman** is university distinguished professor of physics at Kansas State University, where he has been a faculty member for more than 40 years. During his career he has received four major awards*—*the American Association of Physics Teachers’ Oersted Medal (2014), the National Science Foundation Director’s Award for Distinguished Teacher Scholars (2004), the Carnegie Foundation for the Advancement of Teaching Doctoral University Professor of the Year (1996), and AAPT’s Robert A. Millikan Medal (1995). His present research concentrates on the teaching and learning of physics and on science teacher preparation.

**Previously**

What Are Things Made of? Depends on When You Ask.

Ancient Greeks Were the First to Hypothesize Atoms

The Poetry of Atoms

Atom Theory in Ancient India

Religion, Science Clashed over Atoms

Medieval Arabic Scholarship Might Have Preserved Scientific Knowledge

Rediscovering a Roman Poet – and Atom Theory – Centuries Later

Reconciling Atom Theory with Religion

Did Atom Theory Play a Role in Galileo’s Trouble with the Inquisition?

Did Gifted Scientist’s Belief in Atoms Led to His Obscurity?

Does Atom Theory Apply to the Earthly and the Divine?

A Duchess Inspired by Atoms

Separating Atoms from Atheism

Isaac Newton: 300 Years Ahead of His Time

Issac Newton and the Philosopher’s Stone

When Chemistry and Physics Split

Redefining Elements

Mme Lavoisier: Partner in Science, Partner in Life

With Atoms, Proportionality and Simplicity Rule

Despite Evidence of Atoms, 19th Century Skeptics Didn’t Budge

Mission of the First International Scientific Conference: Clear up Confusion

Rivalry over the First Periodic Table

The Puzzle of Dark Lines amid Rainbow Colors

The Colorful Signature of Each Element

Light Waves by the Numbers

Even Scientific Dead Ends Can Contribute to Knowledge

Discovery of the Electron Took Decades and Multiple Scientists

‘Wonders of the X-ray’

The Accidental Discovery of Radioactivity

Marie Curie: A Determined Scientist

Pierre and Marie Curie Extract Radium – and Pay a High Price

Scientists Delve into Radioactivity and Make Their Own Discoveries

The First Attempts to Visualize Atoms

Did Busy Work Lead to Models for Atoms?

Why Does Ice Melt? The Answer Lies in Physics.

Einstein Explains How a Dim Light Can Release More Energy Than a Bright One

How Bohr’s Famous Model of the Atom Was Created

Bohr’s Model of the Atom Answers Fundamental Questions – but Raises More

Bohr’s Model of the Atom Draws Critics

‘A First Feeble Ray of Light’ to Explain Electrons’ Orbits

Two Labs across the Atlantic Prove That Electrons Behave Like Waves

A Mathematical Approach to Atoms That Works but Is Complex

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