Erwin Schrodinger -- His Life and the Cat:
Erwin Schrodinger w as born on August 12, 1887 in Vienna. It would seem that from an early age he was “destined” to a life of scientific discovery. His mother was the daughter of his father’s chemistry professor at the Technical College of Vienna. His father was well versed in chemistry and took up botany. As a child, Erwin was particularly interested in sciences, ancient grammar, and German poetry. Schrodinger also absolutely hated any form of straight memorization, as well as learning from textbooks.
During Schrodinger’s studies at the University of Vienna, he came under the influence of several important professors and researchers particularly in math and physics. From these professors, Schrodinger gained a significant knowledge of eigenvalue problems in the physics of continuous media. This would be the groundwork necessary for his future discoveries that he is known for today, including the Schrodinger wave equation.
Between the years of 1920 and 1926, Schrodinger took up various positions as an academic researcher and published quite a few papers in a variety of topics. Of these were papers on specific heats of solids, thermodynamic issues, color theory, and atomic spectra. It was not until the beginning of 1926 that Schrodinger began work on his famous wave equation. The Schrodinger Equation relates th
e Hamiltonian (an operator) of the wave function to the energy of the wave.
Ĥ is the Hamiltonian, E represents the energy, and Ψ is the Wave Equation.
Schrodinger had a close personal correspondence with Albert Einstein and often had discussions with him regarding theoretical physics. In 1935, after one such correspondence with Einstein, he proposed the famous Schrodinger’s cat thought experiment, believed to be greatly influenced by Einstein.
Introduction to the Cat Thought Experiment:
S
chrodinger’s thought experiment was set up in the following way. It highlighted principles dealt with in the Copenhagen interpretation as well as other quantum mechanical models. The particular principal that the thought experiment was dealing with was entanglement and that the cat could not be determined to be dead or alive.
The “experiment” (as it wasn’t actually executed) was set up as follows: A cat was placed in a sealed steel chamber along with some radioactive material, and a Geiger counter. When (or if) the radioactive material decayed within an hour (the amount of radioactive material was set so it was a 50/50 for this to occur) the Geiger counter would detect the decay and trigger a hammer that would smash a flask of a toxin killing the cat. Because the radioactive decay of the material is governed by a wave function, it is uncertain as to whether the cat is alive without observation.
The Heisenberg Uncertainty Principle:
I
n 1927, the physicist Werner Heisenberg published a paper on something that turned the physics world upside down. The general idea was simple: one could not measure both the position and momentum of a wave-particle at the same time with exact certainty. This was expressed in an experimentally determined mathematical expression, where ∆x is the uncertainty in distance, ∆p is the uncertainty in momentum, and ћ is Planck’s constant divided by 4π (approximately 1.05457E-34 m2kg/s). This equation could also be expressed in terms of ∆E, energy, and ∆t, time.
This concept can be more easily visualized by the graphs below. (We assume that uncertainties take the shape of Gaussian distributions). As you can see, when the uncertainty in momentum decreases, the uncertainty in distance increases; in the same way, when the uncertainty in distance decreases, the uncertainty in moment increases.
Much of quantum mechanics is based on this principle, and thus particles are described using wave functions that predict probability. In Schrodinger’s thought experiment, this uncertainty is what makes it impossible to determine whether the cat is alive or dead, because of the wave function of radioactive particles.
The Copenhagen interpretation:
The Copenhagen interpretation is not one single statement or a law of physics. It is actually a whole understanding of quantum mechanics itself. Much of the quantum mechanics taught in CHEM 260 falls into the scope of the Copenhagen interpretation: it is essentially the expression of many basic ideas such as wave-particle duality, the wave function, and uncertainty. It is basically a conglomeration of much of the commonly accepted modern quantum mechanical laws. It has been an interpretation that has therefore been in development since the dawn of the quantum era, perhaps around the start of the twentieth century. It involves scientists as early as Bohr and Planck.
One of the most important aspects of the Copenhagen interpretation is the idea of the wave function. It defines the whole view of matter leading to quantum theory. The wave function is an abstract solution to Schrodinger’s equation that describes any matter wave. Copenhagen’s interpretation states that all matter exists in multiple probabilities, and those states’ respective probabilities are all described by their wave function. The wave function therefore serves as a sort of position indicator for matter. Most other Copenhagen interpretation ideas stem from the idea of the wave function and probability.
Of the many offshoots of this idea, the most important in the Copenhagen interpretation is wave function collapse. The original argument about the wave function is that while we can be certain that the state of matter which we observe in the real world (as it would seem to be following the laws of classical physics), there is still just as much probability for the matter to exist in all other states permissible by the wave function. In mathematical terms, it is important to understand that each wave function is actually a sum of multiple eigenfunctions1, each one representing a possibility for that wave function, or a possible state for that matter. Just as squaring the wave function gives the probability of the particle being in that respective state, the sum of all these eigenfunctions squared must give the sum of all probabilities for the wave function: this must be one for a particle to exist. This is said to be “normalized”. Wave function collapse states that when we observe a specific state in the real world, we have pinpointed the specific eigenfunction that that particle exists in. The wave function thus collapses to this single eigenfunction. That is to say, in quantum theory once we see a particular state we theoretically ignore all other probabilities for that particle, since we know what state it is in. It is interesting to note that the many world theory states that while it may be known by the observer what state the matter is in, all other probabilities originally determined by the wave function are said to occur in parallel universes. In those universes, their wave functions also collapse into their respective states.
Heisenburg’s uncertainty principle is also accepted in the Copenhagen interpretation. This is a central principle in quantum mechanics that states that only certain measurable quantities can be known with precision, but it would cause other quantities to only be describable by probabilities. Some other ideas that Copenhagen interpretation encompasses are the concepts of measuring: that measuring devices can only be used to measure classical devices. However, by the correspondence principle, classical mechanics still apply to particles of appreciable size, and quantum mechanics are still mostly accurate for such particles.
Complementarity, the idea that all matter exists as both waves and particles, is also widely accepted among “Copenhagenists”. It follows the concepts of uncertainty in that complementarity treats classically intertwined values such as space/time and energy/momentum as separate values. Such things cannot be viewed at the same time, such as position and momentum under Heisenberg. It is stated that all matter therefore has two states: wave and particle, which are two separate aspects of that matter.
The Copenhagen interpretation is the encompassment of all the laws of physics that led to the proposal of Schrodinger’s cat. He posed the question in an attempt to put the laws of uncertainty into a larger scale: a cat instead of, suppose, an electron. Under the Copenhagen interpretation, the cat can only be in two states, and is therefore 50% alive and 50% dead. Wave function collapse dictates that once the box is opened up and the state of the cat is determined, the wave function collapses on the one state determined, and that in other worlds there exists a vice-versa state, that is, if a dead cat were discovered, in 50% of the other worlds the cat is actually alive.
1Eigenfunctions are functions that, when put through an operator (such as multiplication, addition, division, differentiation, etc...), returns the original function multiplied by a numerical value.
References:
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"Copenhagen Interpretation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)." Stanford Encyclopedia of Philosophy. Web. 08 Oct. 2010. <http://plato.stanford.edu/entries/qm-copenhagen/>.
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Oxtoby, David. Principles of Modern Chemistry. Vol. 1. Mason, Ohio: Cengage Learning, 2008. Print.
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