Ron’s Pre-discussion Primer: From Newton’s Stable Universe to Schrodinger’s Cat

February book discussion: The Quantum Moment (2014) by Robert P. Crease and Alfred Scharff Goldhaber

 

Newtonian Physics

Newton’s Principia, published in 1687, described a stable and predictable universe:

  • Space and time are both treated as universal constants that have the same properties wherever or however they are observed and measured.
  • Space stretches out in all directions to infinity, and time goes on forever from an infinite past towards an infinite future.
  • Time and space are both continuous with no gaps or discontinuities.
  • Within these dimensions of time and space, objects are acted upon by forces that cause them to move and change in ways that can be explained (and, at least in principle, predicted) based solely on deterministic mechanisms (no need to invoke interventions of Gods or Spirits or other unknown elements).
  • Although not spelled out explicitly because it was taken for granted, there is embedded within Newtonian physics an implicit assumption that there is an objective physical universe that exists independently of ourselves, and that an ultimate Theory of Physics should be able to describe all of the properties of that universe.

Newtonian physics still works adequately for scientific and engineering purposes as long as the objects being described are medium sized and moving at moderate speeds. However, predictions about what is expected to happen based on Newtonian physics break down when applied to objects that are extremely small, extremely large, or moving very fast. Under those conditions every one of the bullet point assumptions shown above fall apart. In the following sections, I try to provide a brief historical summary of how this came about. I have (over)simplified many concepts in order to try to optimize pedagogical clarity, but I think the main thrust of this narrative is accurate.

Einstein’s Theory of Special Relativity

The first chink in the armor of Newtonian physics was the assumption that space and time are the same wherever or however they are observed and measured, This assumption was jettisoned by Einstein when he published his Theory of Special Relativity in 1905. His theory can be summarized in a nutshell by one of the most famous equations in the history of science,

E = MC2

where E is the amount of energy, M is the amount of mass, and C is the speed of light in a vacuum. Note some interesting aspects of this equation. First, energy and mass are not qualitatively different things. Mass can be changed into energy (and vice versa) and when that happens the results will be as shown in the equation. This was the theoretical basis that provided the rationale for building the first atomic bomb – The tiny amounts of mass that are present in atoms are enough to produce huge amounts of energy due to the fact that in the transformation the mass will be multiplied by the speed of light squared.

Another interesting thing about this equation is that while E amd M are variables that are allowed to take on different values, C is a constant that is never allowed to change. Einstein deliberately structured his theory such that the speed of light in a vacuum must remain constant throughout the physical universe.

The variables M and E are shorthand notations for mathematical expressions made up of many smaller terms. By the time M is fully expanded it will be a complicated set of equations that fill an entire blackboard. Same for E. The expanded forms of the equations have variables in them such as meters/second. And this is where we get the dramatic predictions made by Einstein’s theory. In order to get E = MC2 to work mathematically, terms like meters and seconds have to be variables that take on different values depending on the conditions under which they are measured. In other words, in Einstein’s theory about how the physical universe operates, rulers have to shrink or elongate and clocks have to speed up or slow down, depending on the conditions under which measurements are made. Most “ordinary” people would have have considered this notion too crazy to possibly be true, and would have discarded it. But Einstein was not an ordinary person. He decided that these mathematical equations really were accurate descriptions of the way the physical world works.

For most kinds of measurements we might make about properties of the physical universe, Newton and Einstein make predictions that are not discernibly different. However under conditions where objects are traveling at high velocities, the two theories make different predictions, and in these cases it has been demonstrated repeatedly that Einstein’s theory always wins. Clocks really do speed up or slow down and rulers really do shrink or elongate by amounts large enough to be noticeable. An example of a modern-day practical application where these differences are large enough to matter is GPS devices. If the position measurements based on a GPS device were not corrected for the effects of relativity, they would accrue an error of about 10 kilometers within the first day the GPS satellite was in operation, and would continue to get steadily worse as time goes on!

We can eliminate the first bullet point:
Space and time are both treated as universal constants that have the same properties wherever or however they are observed and measured.

Einstein’s Theory of General Relativity

Einsteins Theory of Special Relativity was only designed to apply under conditions where the effects of acceleration and/or gravity could be ignored. Einstein immediately went to work on a more general theory, and this was completed a decade later in 1915. Its equations describe a universe that is finite in size. Furthermore it is increasing in size over time. Einstein was so troubled by this latter aspect of his theory that he added a constant that had only one purpose – to keep the size of the universe constant over time. It is ironic that this same person who 10 years earlier had fearlessly followed the implications of his equations even when they allowed Newtonian space and time to warp, had now lost the courage to accept the implications of his newer theory. He stated later that adding this constant was the biggest blunder of his scientific career.

Other physicists and mathematicians who examined his theory later ignored Einstein’s caution and examined the effects of decreasing the value of the variable in the equations that specified time (in other words, modeling the effects of letting time run backwards). When that operation is carried out, the equations reveal that the size of the universe shrinks to zero about 14 billion years ago. This moment in time before which our universe did not exist is now referred to as The Big Bang.

Numerous empirical measurements made by astronomers over the last 100 years are consistent with Einstein’s theory rather than Newton’s. Newton’s infinite universe has been superseded by one that is expanding but finite in size and that has only been in existence since a big bang occurred about 14 billion years ago so we can eliminate the second bullet point:
Space stretches out in all directions to infinity, and time goes on forever from an infinite past towards an infinite future.

Quanta

In 1900 Max Plank was trying to use mathematical equations to model the observation that as objects are heated they change color, becoming first red and then shifting to a bluish-white. Plank discovered that he could only model these changes if he incorporated a constant into his equations, now commonly referred to as the Plank Constant. Plank just considered this to be a mathematical convenience that was needed to make his equations work, and did not posit any physical meaning to it.

In 1905, Einstein was working on a related problem that involved interactions of light with metals. He incorporated the Plank Constant into his equations, but took the additional step of asserting that this constant referenced an actual physical property of light. (Note: Einstein is mostly remembered by the general public for his theories of relativity. However, his Nobel Prize was actually for his introduction of the quantum into physics, a discovery that launched Quantum Theory, the implications of which Einstein himself was never comfortable with.)

Einstein’s light quanta, later renamed photons, are discrete packets of light energy. Prior to Einstein, light had always been considered to be a continuous wave that travels through space and can be composed of any arbitrary amount of energy. Einstein demonstrated that this conception is wrong (or more precisely, is inadequate as we will discuss below). Measurements of light intensity show that it is made up of individual photons, so while it is possible for example to have one photon, or two photons, or a trillion photons, it is never possible to have a fraction of a photon.

This initial discovery of the quantal nature of light soon spread to other aspects of physics. Bohr demonstrated that orbits of electrons around nuclei in atoms can only occur at discrete energy levels related to Plank’s Constant, i.e., that they are quantized. Soon other examples of quanta started showing up ubiquitously in other contexts of physics. Even time and space are quantized when examined at their grainiest levels, analogous to the way a moving image on a television screen appears continuous when viewed at a normal viewing distance, but is actually composed of a number of very small spatially separated pixels whose intensities are changed at discrete time intervals.

Time does not exist in intervals smaller than Plank’s Time Constant, nor space that occupies an area smaller than Plank’s Space Constant. Furthermore, changes in time and space that occur at these granular levels of resolution are not continuous. They involve instantaneous jumps from one discrete location to another.

A third bullet point bites the dust:
Time and space are both continuous with no gaps or discontinuities.

Unpredictability

Another implication of Quantum Theory, as it was further developed in the first half of the 20th Century, is that it is not possible to give a deterministic prediction about what will happen to a very small particle in the next moment, even if everything about that particle’s prior history and the forces currently acting on it are specified. Quantum theory asserts that the most we can ever hope to attain, based on first principles, is a list of the probabilities specifying, for example, the likelihood that a particle will appear at a various location in space at the next moment. This was the basis for Einstein’s famous rejection of Quantum Theory summarized in his playful quote, “God does not play dice!” However, an increasing amount of experimental evidence obtained from studies performed both during Eienstein’s lifetime and since his death has consistently come down on the side of supporting Quantum Theory so lets cross off another bullet point:

Within these dimensions of time and space, objects are acted upon by forces that cause them to move and change in ways that can be explained (and, at least in principle, predicted) based solely on deterministic mechanisms (no need to invoke interventions of Gods or Spirits or other unknown elements).

Or at the least, replace “deterministic mechanisms” with “probabilistically determined mechanisms.”

Uncertainty Principle

Worse yet, Heisenberg, with his Uncertainty Principle, demonstrated that there are some aspects of the physical world that are, in principle, unknowable. Whenever we make an observation of the physical world, that observation itself might alter properties of what is being observed. It had always been assumed by Newton and the physicists who followed, including Einstein, that Theories of Physics describe properties of an objective world that exists independent of ourselves. Quantum Theory has had to abandon that assumption. It simply provides a set of probabilistic statements about how likely it is that specific outcomes will occur if we decide to make a specified set of observations of the physical world. It totally gives up, based on first principles, on the quest of trying to discover properties of the physical world as it might exist independent of our observations.

This eliminates our last remaining bullet point:
Although not spelled out explicitly because it was taken for granted, there is embedded within Newtonian physics an implicit assumption that there is an objective physical universe that exists independently of ourselves, and that an ultimate Theory of Physics will eventually be able to describe all of the properties of that universe.

Double Slit Experiment Weirdness

I described earlier how Einstein demonstrated that light has a quantum nature when its intensity is measured. However, light also has a wave nature that can be revealed when other kinds of measurements are made. The term wave-particle duality is used to refer to the fact that light has some properties that are only consistent with it being a discrete particle and other properties that are only consistent with it being a wave. And some of these two sets of properties appear to be incompatible with one another. This situation where two statements are both considered to be true and yet appear to be inconsistent with one another is referred to as a paradox. Apparent paradoxes are another ubiquitous, yet puzzling, feature of Quantum Theories.

A prototypical quantum physics paradox involves double slit experiments. These were initially performed with light, using the following set of procedures. A beam of light is aimed towards an opaque barrier that has two narrow slits, positioned side by side, that allow light to pass through. If a screen is set up on the other side of the barrier, a series of stripes can be seen that is referred to as an interference pattern.

The appearance of an interference pattern on the screen was traditionally considered to be evidence that light is a wave. It is easy to visualize the logic for this interpretation by thinking about an analogous situation involving waves in a pond of water. Imagine that a pond has a barrier running down the middle with two narrow openings (slits) allowing waves to pass from one side of the pond to the other. Now imagine that a large rock is thrown into the pond on one side of the barrier. Waves will spread out from where the rock lands and travel out in all directions, eventually hitting the barrier where only the portions of the wave that pass through the two slits will make it to the other side of the pond. As the waves come out of the slits on the other side of the barrier they will fan out in all directions and interact with one another. In some locations the peaks of two waves, one from each slit, will arrive at the same time, adding to one another to create a new wave with a peak twice as large as either wave alone. At other locations, two troughs will arrive at the same time creating a larger trough, at other locations various combinations of peaks and troughs will interact, etc. By taking into account all of these interactions, it is possible to predict what the new wave will look like when it hits the shore on the other side of the barrier. Similarly with light, it is possible to calculate all of the expected interactions as the waves fan out from the two slits and interact with one another before they land on the screen behind the barrier. The expected result is the exact interference pattern that is in fact seen on the screen when a double slit experiment is performed.

This result seems pretty straight forward, but some results that appear paradoxical are obtained when this experiment is modified in certain ways. First, the intensity of the beam of light that is directed at the slits can be reduced low enough to guarantee that only one photon will arrive at the slits at the same time. The screen on the other side of the slits is replaced with a photographic film that will be exposed as each photon arrives. When this experiment is performed and the film subsequently developed, an interference pattern is seen! This seems to imply that each individual photon somehow passes through both slits and interferes with itself.

Next consider the following experiment. Instead of placing a photographic film, a telescope with a photodetector attached to its viewfinder can be placed behind the screen and aimed such that it can only capture light arriving from one slit or the other but not both. If we run this experiment, 100% of the expected photons will be detected by the telescope. It “seems as though” somehow a photon, when it arrives at the slits, “knows what is behind the screen”, and if it is a photographic plate, the photon passes through both slits simultaneous to create interference, but if it is a telescope aimed at only one slit, it passes only through the slit that is being observed. (This is not the only possible interpretation of these results, but it would take a considerable amount of additional discussion to deal adequately with these issues. This limited description should be sufficient to illustrate why these results are considered to be paradoxical by many.)

Even weirder, are the results from a delayed double slit experiment first proposed by theoretical physicist John Wheeler. Using very fast and sophisticated equipment, a time can be specified when the photon had to have already passed through the slit(s), and only after that time is either a telescope or a photographic plate moved into place to detect the photon. The results are the same as for the standard double slit experiment. “Seemingly” (at least according to some interpretations) a photon is able to detect, after it has already passed through the slit(s), what is now revealed to be positioned on the other side of the screen, then back up in time and retroactively cause itself to have passed through whichever slit(s) is now appropriate for that observing instrument!

These results are non-intuitive, but the mathematical equations used by Quantum Theory accurately predict these outcomes. So Quantum Theory works, but its predictions are probabilistic, and in many cases appear to go counter to “common sense.”

Schrodinger’s Cat and Beyond

The results discussed in the last section involve experiments in which light energy passes through double slits. In the 1920’s, Schrodinger and others developed Quantum Mechanics, wave equations that apply not only to photons of light, but to all particles including those that have mass.

Quantum Mechanic predictions seem to apply only to small objects (subatomic particles, atoms, and small molecules). Larger objects, such as humans and cats, appear to behave according to classical Newtonian and/or Relativity principles. But where is the dividing line? It is not exactly clear.

It is possible to throw small bits of mass through a double slit and the results reveal, surprisingly, a wave-particle duality, the same as for light energy. Recent double slit experiments show that wave-particle duality can be demonstrated routinely for buckyball molecules (named after Buckminster Fuller) that are composed of about 60 atoms, and the current record is for a molecule made up of over 10,000 atomic mass units, a large enough piece of matter to be visualized in a microscope.

This brings us to a tongue-in-cheek “thought experiment” proposed by Schrodinger about a cat. He imagined that a cat is inside a box under conditions such that it might be dead or might be alive. Schrodinger asked, Is the cat dead or alive before we open the box to observe it? This is analogous to the double slit experiment where it can be asked, Is the object passing through the left slit, the right slit, or both slits? The answer provided by Quantum physics is that the object “did all three” (or perhaps equivalently, “did none of the above”). Only after the object is later observed does it take on a reality with respect to these questions. If we were to extend Quantum theory from small particles to larger objects like cats, we would have to conclude, similarly, that the cat is [both/neither] alive [and/nor] dead until it is observed.

Those of you who have read our supplemental reading for this month, Trespassing on Einstein’s Lawn, will have encountered an attempt by some current theoretical physicists to apply wave-particle duality to objects as big as the entire universe. Current theories include the idea that our universe is not the only one that exists, but that it is one of many universes that collectively form a multiverse. A question that has been raised by this idea is, How does it happen that we are living in this universe and not some other? One proposed answer is that all possible universes form a wave function, defined only in terms of probabilities but not in terms of existence. A particular universe comes into existence only when it is observed. According to that perspective, the reason we live in this universe is because we observed it and that fact created its existence.

Ron Boothe
psyrgb@emory.edu

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About Ron Boothe

I am a retired professor of psychology living in Tacoma Washington USA.
This entry was posted in 2016 Selections, The Quantum Moment and tagged , , , . Bookmark the permalink.

5 Responses to Ron’s Pre-discussion Primer: From Newton’s Stable Universe to Schrodinger’s Cat

  1. Ron Boothe says:

    I have been discussing Einstein’s theory of Special Relativity with my grandson in conjunction with a science project he is doing for his Middle School. Starting at a young age, Einstein himself carried out gedanken experiments in which he thought about what it would be like to try to measure the speed of light. I have discovered that many of Einstein’s concepts can be made accessible to a Middle Schooler via gedanken experiments similar to those Einstein himself carried out. Here is one that provides an intuitive feeling for why Einstein’s theory requires rulers to elongate and shrink and clocks to speed up and slow down.
    ————–
    Suppose you are traveling through outer space inside a box. You are weightless so know that you are going in a straight line at a constant velocity. There are two holes in the walls of the box, one in the front and one in the back. They are positioned such that an arrow pushed through these two holes and extending out the front of the box would aim in the direction of motion along what we will call the x-axis. The only measuring devices you have available are a stopwatch and a ruler. Your job is to measure the velocity along the X axis of objects that pass through the box.

    An object (perhaps a rock with some identifying marks on it) enters through the back and exits out the front. Using your stopwatch you measure that it took 1 sec for the rock to pass through the box, and using the ruler you measure that the distance between the two holes in the box is 1 meter. You calculate a velocity of 1m/sec.

    Later while you are asleep, and unaware, the box is accelerated and positioned in front of the same rock but now the box is traveling at a faster velocity. Next morning you see the same rock pass through the cabin and make the same measurements, but this time you measure 2 seconds and 1 meter. You calculate a velocity of 0.5m/sec. This is repeated several times, and each time the measured velocity of the rock (meter/sec) appears to be slower.

    Now you try to think about how to interpret these results. Three possibilities come to mind.
    1. The rock is slowing down.
    2. Your box is speeding up.
    3. The ruler you used to make the measurements elongated and/or your clock slowed down.

    Later, you are able to look at the logbook of what was happening to the box while you were asleep and can now calculate velocities for the box based on the accelerations applied. Using this new information, you discover that if you now add a correction to your measured results,

    [Corrected rock velocity] = [measured velocity] + [increased velocity of the box]

    everything seems to make good common sense. Your original measurements seemed to violate Newton’s laws — a rock traveling unimpeded through space should not change its velocity. Your corrected results save Newton’s law. Furthermore there is no need to assume something crazy like elongating rulers or slowing down clocks. You decide that from now on, when you make measurements from inside this box, you should always correct the measured results based on the velocity of the box.

    Now you repeat the same experiment except this time instead of a rock passing through the box, you measure the velocity of waves of light as they pass through the box. This time you obtain a rather remarkable result that is different from what happened with the rock experiment. The measured velocity of light waves stays the same regardless of the velocity of the box. So if you try to apply the correction that takes into account the velocity of the box, it has the effect of making the speed of light change. In your physics courses you have learned that Maxwell’s equations state that the speed of light is constant.

    You try to think about how you could interpret these new results in a way that would be consistent with Maxwell’s equations. The only thing you can come up with is the number 3 possibility that you had considered, but rejected, when you did your original rock experiments — Somehow the ruler elongated/shrunk [and/or] the clock speeded/slowed in just the right amounts to keep the measurements of light velocity (distance/second) the same under all conditions.

    You go to work to see if you can find some transformations of space-time that would have just exactly that effect. Turns out the Lorentz transformations will do it. You adjust the values for the measured velocity of light and of the box in your equations. Your times in seconds (t) are Lorentz transformed to values of t-prime and your values distances in meters (m) to Lorentz transformed values of m-prime, and Voila, everything works. Your (transformed) values (m-prime/t-prime) now all give the same result for the speed of light, regardless of the velocity of the box, and they also work for your original rock experiments.

    You have to decide what is more important, keeping the speed of light constant or common sense. If you are Einstein, you sacrifice common sense and build a Theory of Special Relativity that allows clocks to speed and slow down and rulers to elongate and shrink.
    —————————-

    One limitation of this gedanken experiment is that it does not provide any intuitive insight about how this transformation of space-time coordinates would have anything to do with relationships between energy and matter, as reflected in Einstein’s famous Special Theory of Relativity equation,
    E = mc^2
    Since I do not have enough mathematical background to work through Einstein’s equations, I have always just had to accept as a matter of faith that somehow when you make these adjustments to space-time within Maxwell’s equations, E=mc^2 pops out.
    Ron

    • Ron Boothe says:

      Richard Smaby sent me the following YouTube links. The first presents essentially the same gedanken experiment I described, but in a somewhat more sophisticated manner:

      The second illustrates how to derive the Lorentz transformations of time and space that are necessary to keep the speed of light constant:

      The following 2 YouTube lectures in this series, Lecture 3 and Lecture 4, elaborate in more detail some of the implications of these transformations.

      The final lecture in this series, Lecture 5, gives a good explanation of how the Lorentz transformation relates to Einstein’s e=mc^2 equation.

      All of the equations in these videos should be able to be followed, with a moderate effort, by anyone with no more mathematical background than Advanced Algebra. I learned a lot by watching these!

      Thank you Richard for bringing these to my attention.
      Ron

  2. Richard Smaby says:

    A couple more links regarding Einstein’s derivation of e=mc^2

    First a ‘thought experiment’

    Next a mathematical derivation

  3. Mohsen Mirghanbari says:

    “The Quantum Moment”

    Ron, I think your book selection “Quantum Moment” deserved an applaud, a reading that required greater understanding, I had hoped for a broader debate and expected wider learning spectrum and a bit of scientific debate if you would, after all with names like Einstein and Newton nothing less should have been the rhetoric.

    Whether, one possess the Newtonian or the Einsteinium Quantum Physics understanding or the higher Mathematical values is irrelevant, however, simply put now-a-days, our environment is filled with gadgets and instruments so different than that of any other time in mankind’s history, thus, requiring a different form of learning, perhaps a new Quantum Physics separated from that of that taught in traditional education.

    Perhaps Einstein’s greatest contribution to the modern man was not the understanding of Earth’s graviton sciences, rather relativity of knowledge itself.

    Listening to Richard S. express his viewpoints during our discussion, so unobtrusive and discreet, while sharing valuable knowledge was very inspirational.

    I came away with both questions and answers, which is a sign of a good read. Questions such as, was Einstein born a genius with an upper echelon IQ’s or did his spontaneous interest allow him to further pursuit his interest? Would he have discovered his gravitational Relativity theory without his predecessor’s findings?

    Einstein’s theories of action at distance and the concept of acceleration or motion, the electrodynamics of light signals, and moving clocks or gravity equals accelerations, and the space curvature (Geodesics), as well as Relativity (gravitational-mass is the same as inertial mass), allowed me to rethink the “Live load/ Dead Load” calculative numbers used in producing structural design strength values.

    Of the Newtonian “Motion” theory, that all objects smaller and larger require both distance and time before reaching peak speed and depend on mass weight and force, that all illumine systems (light source) and energy forces lose velocity with distance, thus require rebooting (reenergizing), similar to that of an engine which requires time and distance for maximum thrust.

    Among the many questions, the nucleus of most scientific research is based on the un-observed, that the “un-observed is both there and not” and if so, then “there must be a God” even though it is un-observed, right?

    And “In this age of populism” the impossibility versus the indeterminacy, whether we have a free will, that all theories hold that we do not, all experiences that we do, however, a similar paradox occurs in the realm of impossibility theory, Many things occur in biology that all science says they are possible, while all common sense says they are not.

    Common sense says that it exist, while the science says it does not, and the fact that it does not lend a certain weight to the argument, each step in this process can be shown to follow the laws of Chemistry and Physics. It all works, there is no need for spirits or poltergeists to explain it, except that it obviously does not.

    “the un-fathomable technical jargon that is opaque” A useful trait of human mind, one that keeps us from going mad, is that if we see something enough times we begin to believe that it make sense. This is not just the basis of Quantum Physics, but most of life.

    While this does not seem likely at all, it does fit nicely into dominant paradigm crossword puzzle and tinker toy, this holds that the Universe is like a crossword, some parts we have filled in, others we have not figured out yet, but we know that is only matter of time until we have it all. Nothing is inherently unknowable, We can fill in the puzzle.

    While I have come to learn and appreciate so much of Einstein’s extraterrestrial scientific findings, however, I also reserve the right to better understand his personal thinking values, it’s important to me, because his scientific research and findings have impacted many lives, yesterday, today and will continue to alter more lives tomorrow.

    Mohsen

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