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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

Dean ZollmanAs 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

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

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.

Schroedinger on the Austrian 1000 shilling note

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


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.)

Shroedinger equation

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.


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 waysfrom 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 awardsthe 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.


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