Schrödinger: the man, his method, and his cat
I've been busy lately working on this semester's term paper on Nuclear Waste Management, so in order to maintain activity on my blog, here's last semester's term paper regarding the life and works of Erwin Schrödinger.
“Even if I should be right in this, I do not know whether my way of approach is really the best and simplest. But, in short, it was mine. The 'naïve physicist' was myself. And I could not find any better or clearer way towards the goal than my own crooked one.”
Erwin Schrodinger, What is Life?
Erwin Rudolf Josef Alexander Schrödinger (12th August 1887 - 4th January 1961) was one of the defining authors of our modern understanding of quantum chemistry. Schrödinger's work, via the application of eigenfunctions and eigenvalues to atomic and molecular species via wave mechanics, applied a successful mathematical model that, to this day, has been celebrated as one of the crowning achievements of the 20th Century. In an era defined by chaos, his ability to pioneer through such times of both personal and global turmoil gave humanity an original, yet revolutionary, way of thinking when it came to the perception of physical and chemical mechanics at the quantum level. This literature work will delve into the history, progression, and culmination of Schrödinger's life and research as we discuss, within the scope of Erwin Schrödinger himself, the man, his method, and his cat.
Schrödinger was born August 12th, 1887, in Vienna, Austria, as the son of botanist Rudolf Schrödinger, and grandson of one of his future chemistry instructors, Alexander Bauer. He was raised in an environment saturated with intellectualism. As he progressed through his education, Schrödinger showed a remarkable ability to quickly understand the nuances of lecture material in real time and apply such material with confidence and ease. Schrödinger studied physics, culminating in the receipt of his Ph.D. at the University of Vienna in 1910. Upon receiving his doctorate, he accepted a faculty position in physics at the University of Zurich.
“… we all knew that he took in everything during the instruction, understood everything, he was not a grind or a swot. Especially in physics and mathematics, Schrödinger had a gift for understanding that allowed him, without any homework, immediately and directly to comprehend all the material during class hours and to apply it. After the lecture of our professor Neumann who taught both subjects during the last three Gymnasium years it was possible for him to call Schrödinger immediately to the blackboard and to set him problems, which he solved with playful facility.” Walter Moore - Schrödinger, Life and Thought
Upon his arrival at the University of Zurich, Schrödinger came upon the work of Louis De Broglie, who was proposing the use of wave mechanics in understanding atomic species. Using such direction, Schrödinger published a series of papers in 1926 titled “Quantization as a Problem of Proper Values” that explored the possibility of characterizing atomic and molecular species via a relationship between wave functions and their corresponding eigenvalues. His derivation, beginning with the Hamilton-Jacobi differential equation:
Sets q as the independent variable, and S as an unknown product of related functions of the single coordinates. Taking H as a Hamiltonian function for Kepler motion, it's shown that, for negative values of E, a discrete set of proper values emerges. These negative values correspond to the Balmer Series for a hydrogen atom. For numerical agreement between the derivation and the Balmer Series, K is equated to a value of h/2π. The above equation transforms to the following:
and via integrating over all space, the above equation returns:
In his original publication, Schrödinger continues from these conclusions and redefines his formula for polar coordinates:
from such equations the Bohr Energy levels are obtained, with L being the principal quantum number:
The derivation in full can be found in full in Schrödinger's 1926 paper, "Quantisation as a Problem of Proper Values". It was for this breakthrough that Schrödinger won his 1933 Nobel Prize. The discovery is, to this day, considered one of the major breakthroughs of the 20th century.
Even though Schrödinger's work is, for our era, considered a breakthrough, there were several characters of Schrödinger's time that found themselves to be at odds with the probabilistic interpretation of the Schrödinger wave interpretation - namely, Schrödinger himself. The Schrödinger Equation was, first and foremost, designed with the intention of modeling the behavior of matter waves from a position of causality. However, fellow physicist Max Born reinterpreted his wavefunction with a probabilistic interpretation, to coincide with Werner Heisenberg's work of indeterminacy. This interpretation of probability and indeterminacy in the wave function, much to Schrödinger's dissatisfaction, became generally accepted amongst the global physics community, earning the title of "The Copenhagen Interpretation". It was in rebuttal of this interpretation that Schrödinger penned the following excerpt in his manuscript titled, "The Present Situation in Quantum Mechanics"
"One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a 'blurred model' for representing reality. In itself it would not embody anything unclear or contradictory." Erwin Schrödinger - The Present Situation of Quantum Mechanics
This probabilistic interpretation of the Schrödinger wave equation was also parodied by fellow Nobel laureate Albert Einstein. The two, who had maintained a well-documented friendship over the years, in the wake of such interpretation, began collaborating on research to form what was known at the time as the unified field theory, which hoped to both merge the fields of quantum mechanics and special relativity, as well as close the holes permitted with the current mathematical interpretations, into one, all-encompassing theorem. However, as time passed, both physicists (more so Schrödinger, however) became augmentedly grandiose in the public eye with their work. They turned from colleagues to adversaries, becoming willing and able to both entertain and return verbal jousts towards each other as they drifted apart in their quest to explain the fundamental laws of nature. Schrödinger himself exclaimed to a fellow physicist John Moffat, "my method is far superior to Albert! Let me explain to you, Moffat, that Albert is an old fool".
In the end, however, no matter how bombastic or flamboyant either figure was in the press with their findings, neither Einstein nor Schrödinger were successful in the formulation of an ultimate theory to explain reality as we know it. However, such failures and trifling words do not dampen nor darken the significant contributions each man made to humanity's understanding of our reality. To this day, Schrödinger's wave equation is the foundation to our quantum mechanical understanding of atomic and molecular species. And although we as a humanity continue to move forward with new ideas and new theories that explain the world around us, it's essential to learn from the men who pioneered the path we walk. As such, it's important to take a moment to recognize Erwin Schrödinger: the man, his method, and his cat.
Appendix 1: In Regards To The Schrödinger Wave Equation
Now, for many students in our current structure of chemistry curriculum (including myself when I started the study of Physical Chemistry in my undergraduate course at the University of Idaho), the Schrödinger wave equation is presented as the foundation of quantum chemical study, with little allude as to how such an equation came about in the first place. In such levels of curriculum, ambiguity can be tolerated due to the course focus being more so towards the direct conclusions and applications in regard to quantum chemical systems, and less so towards the mathematical methods to our madness. Such is the reason I felt it was important to return to the Schrödinger equation's origins to understand where such a partial differential came from. Now that such history has been made familiar, we can begin to understand in regards to the legitimacy of our methodology, as well as where the Schrödinger approach requires variation and perturbation. I hope to demonstrate such details here.
Now, if we have some experience with quantum chemistry, most of us will recognize the following format for the Schrödinger equation:
or when simplified:
where Η is the Hamiltonian. It was mentioned previously that Schrödinger derived his equation with specific reference to the Balmer Series; that being, a simple one-electron system where photons are emitted when said electron transitions from a state of high energy, down to the electronic state defined by the principal quantum number of two. Because his partial differential equation was derived in regards to this two-particle hydrogen-like system, we can take said two-particle system and simplify the incident to exist as two separate one-particle problems. Such problems can be solved exactly via the Schrödinger equation, thus allowing us to confirm the correlation between theoretical and experimental analysis of Schrödinger's wave mechanics in regards to physical realities.
However, physical reality is not solely made up of hydrogen-like systems, and such is where the Schrödinger equation begins to run into problems. For example, let us take a step up from hydrogen and observe what happens to the Hamiltonian in regards to the two-electron system of helium:
where "n" designates regard to the nucleus, e to the electron (1 and 2 respectfully), and r to the radial distances between the various particles (single values representing the distance between an electron and the nucleus, and r sub-12 being the distance between the two electrons).
Even if we evoke the Born-Oppenheimer approximation, we would still need to respect the fact that φ will be a six-dimensional wavefunction, and inter electron repulsion in the Hamiltonian renders the equation inseparable. This issue of inseparability compounds upon itself as one begins to investigate many-electron systems. Because of such inseparability, these higher-electron systems are where we are required to delve into approximation approaches, such as variation and perturbation theory.
Appendix 2: What is Reality?
In understanding the origins of wave mechanics, and how even though there are exact solutions available for specific simple systems, it's important to recognize that most of the time what we're doing is approximating and correlating theoretical mathematics to physical reality. Which begs the question: why in the world do they correlate in the first place? We impose a probabilistic understanding to the building blocks of a reality that, at least at a macroscopic level, is anything but. Why is it that at the end of the day, the mathematics tells us that God plays with dice? Why is it not causastic in the manner that Erwin Schrödinger tried to prove? Or is it? Is our weird understanding a physical reality, or just a figment of our available instrumentation?
The origin of quantum chemistry, and the path our subject has taken from its conception, has been one plagued by consternation and confusion. We see quotes like, "Nobody understands quantum mechanics, they just get use to it" by the Nobel laureate Richard Feynman and, "If you can fathom quantum mechanics without getting dizzy, you don't get it" by fellow Nobel laureate Niels Bohr, and wonder how in the world do we expect to answer with definity the question of causality vs. probability at such a microscopic level. I can't say that I know how we proceed to solve such fundamental questions raised by the mathematical methods that have proved their worth time and time again. Yet, at the end of the day, even though we know not the direction that God, Nature, or Science will take us moving forward, the fact of the matter is that the mere existence of our willingness to question reality shows us that we are not yet done moving forward. There is still more to know, and more to discover. No matter our current thoughts and conclusions, and no matter what past truths we've learned, we should be prepared for what new truths could sill lie ahead. Even as semesters wax and wain; even as years come and go; and even as seasons of life open and close, we can't stop being students - students of a quantum chemical field that is longing for new ways to grow.
1) Schrödinger, Erwin. Collective Papers of Wave Mechanics. Quantisation as a Problem of Proper Values. Parts 1-4. AMS Chelsea Publishing, Providence, RI
2) Schrödinger, Erwin. The Present Situation of Quantum Mechanics. Translated by John D.
Trimmer. Proceedings of the American Philosophical Society. 124, 323-38 (1980)
3) Schrödinger, Erwin. What Is Life? Cambridge U, 1944. Print.
4) W J Moore, Schrödinger : Life and Thought (New York, 1989).
5) Halpern, Paul. Einstein's Dice and Schrödinger's Cat: How Two Great Minds Battled Quantum Randomness to Create a Unified Theory of Physics. New York: Basic, a Member of the Perseus Group, 2015. Print.
6) "Erwin Schrödinger - Biographical". Nobelprize.org. Nobel Media AB 2014. Web. 23 Nov 2016. <http://www.nobelprize.org/nobel_prizes/physics/laureates/1933/schrodinger-bio.html>
7) Brauns, Eric. Physical Chemistry 307 Lecture. Lecture Notes Day 1-26. Presented at the University of Idaho, Spring 2016.
8) Siegfried, Tom. Why quantum mechanics might need an overhaul. Web. 14 Nov 2016. <https://www.sciencenews.org/blog/context/why-quantum-mechanics-might-need-overhaul>
9) Levine, Ira. Quantum Chemistry - Seventh Edition. Upper Saddle River, N.J: Prentice Hall, 2000. Print.
As a final note, I'd like to thank both Dr. Eric Brauns at the University of Idaho, and Dr. Charles Wurrey at the University of Missouri-Kansas City, for being fantastic instructors in regard to my study of quantum chemistry. I would not be the scientist I am today without the inspiration of these two great teachers.