Chapter 16: Schrodinger: The Dead Cat and the Letter ‘i’

Garfield and Schrodinger Erwin Schrodinger was a very unusual man. Born in 1887 in Austria, his father was a successful linoleum entrepreneur while his mother was a stay at home mom, who was half English and half Austrian and was fluent in both English and German. In their household, they used both languages, so when Erwin entered the Gymnasium after being home schooled until the age of 10, he was bilingual. In school, he proved especially adept at physics and math, and was reputed to be able to perform problem calculations on the blackboard, without preparation, utilizing equations he had only just seen in a lecture minutes before.

He received his doctorate in physics in 1910, and then promptly enlisted in the army as an artilleryman, but was able to come back to the University of Vienna as a research assistant in the very next year. He spent the next few years oscillating between the military and his academic endeavors and participated in World War I. He was awarded an assistant professorship in 1914 but continued to support the military until 1918, and was married to Anny Bertel in 1920, the same year that he received a full professorship in Stuttgart. He moved around quite a bit during those early years accepting a position at Breslau, then Zurich in 1921, where he met and befriended Hermann Weyl.

It was there that he began his studies of Quantum Mechanics which culminated in a stunning series of six papers in 1926 in which he outlined the mathematical basis for Quantum Mechanics which remains as its foundation today. Dr. Einstein called him a genius.

He disliked and feared the Nazis and their persecution of the Jews, and in 1933 he surprised the academic world (he was quite famous by this time) by requesting a position outside of Germany, and with the support of Great Britain and the allies, spent that summer in Italy. When he went there, he made a somewhat unusual request to bring along his assistant Arthur March and Arthur’s wife Hilde.

Schrodinger was not a man known for his fidelity. His relationship with his wife Anny seems to have deteriorated shortly after their marriage, at least with regard to sex (although they remained married until his death by tuberculosis in 1961 at the age of 73). He was known to have had a series of affairs, with his wife’s knowledge, that culminated in his relationship with Hilde, who became pregnant during that summer. For her part, Anny, not being satisfied with Erwin, took Erwin’s good friend Weyl as a lover for years. But apparently, Schrodinger was really in love with Hilde, because after that summer when he was applying for a professorship outside of Nazi control, he turned down two distinguished positions, one at Princeton and another at Oxford because he insisted on living with both Anny and Hilde, openly. (What was the problem? Everyone was married!) He received the Nobel Prize in Physics for his work in 1933 which he shared with Paul Dirac. His daughter was born in 1934 and the threesome bounced around the world’s universities for the next few years before settling in at the Advanced Studies in Dublin, Ireland in 1940.

At Dublin, he continued to seduce young women, and is known to have impregnated at least two of them. When he thus isolated from the group of physicists that he was associated with during the 20’s and early 30’s, his work seems to have deteriorated. Although he published other papers, none rose to the level that he had achieved during 1926. He even tended toward the metaphysical, writing a book entitled What is life? wherein he surmised that individual consciousness is but an extension of the unitary, universal consciousness that fills the cosmos (Hoo-boy! Bring on the incense!). Not really the sort of stuff that you expect from a first rate physicist. He is credited with giving Watson the concept for the discovery of DNA, though.

He worked on many topics during that time, and in particular on a Unified Field Theory. And in the process, he came up with a theoretical structure based on Affine Geometry. Not having the benefit of his Quantum Mechanical peers to discuss this with, he went forward with this and wrote a paper titled The Final Affine Laws in 1946 and after it was published stated:

“This is the generalization. Now the Einstein Theory becomes simply a special case… I believe I am right, I shall look an awful fool if I am wrong.”

One should be very careful about one says about his friends and their work. Einstein, who had been a good friend and collaborator up until this point, was not amused and broke off communication with Erwin shortly thereafter. The Final Affine Laws was such a dud that it is difficult to find reference to it today.

Schrodinger is probably best known for the thought experiment that bears his name, that is; Schrodinger’s Cat. It is a quintessential example of the paradoxes created by the modern form of Quantum Mechanics as engendered by Schrodinger, and it goes as follows:

Take a sealed box and put a cat into it (preferably a cat that you are not too attached to, like the neighbor’s tom that always leaves scat in your flower garden). In that container, place a device that contains an effective aerosol poison, such as cyanide (hence the ‘sealed’ part) that will be dispensed if hit by a neutron from a decaying radioactive element that is placed in front of the device’s trigger, like say, a Geiger counter. Once you have waited an appropriate amount of time so that the radioactive element has had a reasonable chance of releasing a neutron that would trigger the release of the poison, you cannot know whether the cat is alive or dead without opening the box to check. Therefore, the cat is both alive and dead until you look.

Uh-huh.

This example is supposed to be a quantum mechanical koan of sorts, and reveal to you a true understanding of the nature of the quantum world, where the outcome is affected substantially by the observer and the act of observing, and the object of the study (the cat) is simultaneously existing in two states, alive and dead, until forced to choose one by being observed.

Seriously, I wonder what he had against cats?

Since he originally penned this experiment, the set up has been transmogrified to utilize a photon having to choose between exhibiting itself as a particle or a wave as the trigger of decision as to whether one causes the death or the is the savior of the cat, but the essence of the choice remains the same. This example also supports the Heisenberg Uncertainty Principle, which we’ll discuss briefly later.

But other than providing a guiltless method for euthanizing cats (Honest, I wasn’t sure it would die!), Schrodinger also gave the world the wave equation that bears his name, and set the path of Quantum Mechanics off the track of certainty and into the woods of probabilistic behavior. That equation looks like this:

For a general quantum system:

Screen Shot 2014-01-26 at 7.05.47 PM

Where

Screen Shot 2014-01-26 at 7.06.00 PM is the wave function, which is the probability amplitude for different configurations of the system.

Screen Shot 2014-01-26 at 7.08.05 PMis Planck’s constant divided by 2π, and it can be set to a value of 1 when using natural units.

 

Screen Shot 2014-01-26 at 7.08.32 PMis the Hamiltonian operator.

 

Of particular significance is the Hamiltonian

Screen Shot 2014-01-26 at 7.10.31 PM

which as an observable that corresponds to the total energy of a particle of mass m in a real potential field V.

What this equation says is: the square root of negative one, multiplied by Planck’s constant times the psi function over time is equal to the total energy of the particle. To utilize this equation, he assumed that the electron could be described as a standing wave function, as suggested by DeBroglie, and he used it to correctly calculate the light energy of a photon emitted by an electron of Hydrogen, in agreement with Bohr. Furthermore, he went on to show that this formula was mathematically equivalent to the other competing descriptions at the time, the matrix method suggested by Heisenberg and Feynman’s path integral formulation. However, he did not know the physical meaning of the psi function. It was Max Born who came to the rescue, interpreting ψ (psi) as a probability function.

Like Dr. Einstein, Schrodinger was never very happy with this interpretation, since it reduced all physical occurrences into probabilistic events. He felt, as did Dr. Einstein (and the author, BTW), that the resort to probability for the calculation of the energy was simply a crutch that supported a body of work that did not fully understand or describe the underlying characteristics of the system.

There are also a couple of other problems with Schrodinger’s equation. First, and this is recognized by all physicists, it is a time independent equation, that is, although it is inextricably linked to time and calculates and energy based on the psi function at any given time, it is based on the assumption of the energy as a standing wave, that is, a wave that does not vary over time. This made it difficult to use in relativistic calculations. It was later modified to a time dependant version, but the underlying assumption, although providing a reasonably good explanation, is essentially divorced from time. Not so good.

The other problem is that the formula contains the forbidden number i, the square root of negative one.

How one can accept the use such a purely fanciful number to determine the nature of fundamental, physical quality that describes an empirically determined system is a mystery to me.

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