Uncertainty – The Key to Quantum Weirdness


All these fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?’ Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.

                       Albert Einstein, 1954

As 1926 was coming to a close, the physics world lauded Erwin Schrodinger and his wave mechanics. Schrodinger’s purely mathematical tool was being used to probe the internal structure of the atom and to provide predictable experimental outcomes. However, some deep questions still remained – primarily with the idea of discontinuous movements of the electron within a hydrogen atom. Niels Bohr, champion of and chief spokesperson for quantum theory, had developed a model of the atom that explained spectral lines. This model required an electron to move to a higher energy level when absorbing a photon, and releasing a photon when it moved to a lower energy level. The point of contention is how the electron was moving. This quantum jumping, as Bohr called it was said to be instantaneous. And this did not sit well with classical minded physicists, including Schrodinger.

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Source: Eureka Physics

At this point in time, Erwin Schrodinger favored the idea of ‘wave packets’, and that matter was simply a concentration of these packets. This allowed for a clearer picture of wave/particle duality. Schrodinger understood that there were many holes in this idea, but he believed they would be worked out over time. In August of 1926, he would meet up with Bohr and a man by the name of Werner Heisenberg in a debate that would send the understanding of quantum theory in a new direction.

Schrodinger would have nothing to do with this “damned quantum jumping” as he would later recall. His classical wave theory had continuity and was easy to visualize. But Bohr and Heisenberg held steadfast in their philosophical view that it was not possible to visualize such quantum phenomenon. The debate raged on, and neither side was able to convince the other. However, all three were moved by the debate…most notably Werner Heisenberg. He had long given up on classical space-time visualizations within the atom and instead relied on theory and results of laboratory experiments. But Schrodinger had forced him to step back and try to visualize at least some atomic aspects. And this would lead him to one of quantum physics’ most important discoveries.

Uncertainty

After the debate, Heisenberg began to ponder the path of an electron through a cloud chamber. This was an obvious way to visualize the position and momentum of an electron. Why could he not visualize such a path of the electron orbiting a hydrogen atom? After exchanging some correspondence with Wolfgang Pauli, he realized that he could reinterpret the square of the electron’s wave function. Instead of giving the probability of being in a specific state, this reinterpretation would give a probability of being in a particular location in the orbit of the atom. Doing so allowed Heisenberg to take Schrodinger’s wave mechanics and his insistence on being able to visualize the innards of atoms and apply his beloved matrix mechanics.

It was Pauli that would make the first connection between the position and momentum of the electron with wave mechanics. Viewing two electron waves on a collision path, one would find that each had a clear position (q) and momentum (p) while they are far apart. But as they come closer together, these values get ‘fuzzy’. He would write:

One may view the world with the p-eye and one may view it with the q-eye, but if one opens both eyes at the same time one becomes crazy.

Heisenberg took Pauli’s work and tried to describe the visible path of an electron in a cloud chamber. He would wrestle with this for a while before he began to ask himself some fundamental questions about what is meant by position. He realized that the position of the electron was only known because of the water droplets that condensed around its path. These droplets are much larger than the electron itself. While one might see the path readily by looking at the entire chamber, what happens when you zoom in? He realized that the position and momentum of the electron could not be known with infinite precision at the same time. Only an approximation could be measured. Heisenberg had discovered the uncertainty principle.

A Fundamental Limit of Measurement

After doing some quick math, Heisenberg found that the product of the uncertainties of an electron’s position and momentum could not be smaller than Planck’s constant. There is a fundamental limit to how accurately you can know the electron’s position and momentum. He illustrated his new findings with a few thought experiments, including the following:

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Source: ChemWiki

Imagine we’re following the electron through our cloud chamber with a high powered microscope, and we wish to measure its position and velocity. The resolution of the microscope will increase with the frequency of radiation. So it would be most prudent to measure the position of the electron with a high frequency gamma ray. The problem is the gamma ray photon will affect the electron’s trajectory, which limits our ability to measure its velocity. The solution is to use lower energy radiation, so that the electron’s trajectory will not be affected by the photon. But lower energy means lower frequency. And because resolution is determined by the frequency, it limits our ability to measure its position. Which brings us back to where we started.

Heisenberg would go on to show that it is fundamentally impossible to observe (or visualize, for that matter) an electron in the orbit of an atom, and that he and Bohr were right all along. His uncertainty principle and its fundamental limitation on measurement would go on to shape quantum theory for the next several decades. It would later be applied to the concepts of energy and time, allowing “virtual” particles to blink in and out of existence, using uncertainty to escape conservation laws. This would become the basis of QED, or quantum electrodynamics.

Author’s Opinion

I’ve read several books about Quantum Theory and even performed some basic experiments. While I will not pretend to understand even a fraction of the rigorous mathematics behind the theory, I can confidently state that anyone with an interest and desire to learn most of the basic aspects of the theory can do so.

One of my gripes with these books is that they tend to start off in Chapter 01 with explaining the concept of wave/particle duality, known as complementarity. This is utterly baffling to anyone new to the theory. How can something that looks like little baseballs be the same thing as what one sees when they toss a pebble into a pond? I would argue that they should first teach the uncertainty principle and focus on the idea of limitation of measurement. It’s not confusing, but fairly straight forward and anyone can grasp the idea without much effort. Once one understands that the location of a moving “tiny baseball” can only be known with a probability less than 100%, it becomes much easier to visualize wave/particle duality. The particle is there and real, just like a little baseball. But because of uncertainty in measurement, it is impossible to actually know it’s there1. We are forced to say that there is a probability it is here, and a probability it is over there. This probability is spread out in the form of a wave. This is why tiny particles display wave-like characteristics when you try to observe (or measure) them. You end up only seeing or measuring this wave of probability, because that’s all that you can see or measure.

So by understanding the above, looking at wave/particle duality is much easier, in my opinion. Understanding uncertainty in measurement is the key to all of the quantum weirdness.

1The question of if a quantum scale object can be considered actually ‘there’, even if it is impossible to know if it’s ‘there’ is a hotly debated subject that continues to this day. Albert Einstein refused to give up causality and held on to his belief that the universe is deterministic in nature (that means the object is really there even though we can’t measure it). While Niels Bohr and Werner Heisenberg pushed the idea that if you can’t measure and know that the object is there, to say that it is there is nonsensical. They believed that on the quantum scale, things like causality and determinism simply cannot exist. I will climb out on a yagi and say that modern quantum mechanics and experiments suggest that Einstein was wrong. It would appear that the quantum tree does not make a sound if no one is there to hear it.

Sources

The Quantum Story, by Jim Baggott. Chapters 9 &10  ISBN-978-0199566846



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