Andrew Zimmerman Jones

Cheat Sheet / Updated 05-10-2024

This Cheat Sheet is intended to supplement Quantum Physics For Dummies, 3rd edition, by Andrew Zimmerman Jones. It begins by reviewing some useful operators used in quantum mechanics calculations. Then it covers a useful method for solving the Schrödinger equation for the quantum wave function, and then how you can use that wave function to calculate probabilities in quantum physics. Finally, it lays out some key moments in one of the most important discoveries of quantum physics: the understanding of wave-particle duality.

Article / Updated 09-14-2023

The multiverse is a theory that suggests our universe is not the only one, and that many universes exist parallel to each other. These distinct universes within the multiverse theory are called parallel universes. A variety of different theories lend themselves to a multiverse viewpoint. Not all physicists really believe that these universes exist. Even fewer believe that it would ever be possible to contact these parallel universes. Following, are descriptions of different levels, or types of parallel universes, scientists have discussed. Level 1: If you go far enough, you’ll get back home The idea of Level 1 parallel universes basically says that space is so big that the rules of probability imply that surely, somewhere else out there, are other planets exactly like Earth. In fact, an infinite universe would have infinitely many planets, and on some of them, the events that play out would be virtually identical to those on our own Earth. We don’t see these other universes because our cosmic vision is limited by the speed of light — the ultimate speed limit. Light started traveling at the moment of the big bang, about 14 billion years ago, and so we can’t see any further than about 14 billion light-years (a bit farther, since space is expanding). This volume of space is called the Hubble volume and represents our observable universe. The existence of Level 1 parallel universes depends on two assumptions: The universe is infinite (or virtually so). Within an infinite universe, every single possible configuration of particles in a Hubble volume takes place multiple times. If Level 1 parallel universes do exist, reaching one is virtually (but not entirely) impossible. For one thing, we wouldn’t know where to look for one because, by definition, a Level 1 parallel universe is so far away that no message can ever get from us to them, or them to us. (Remember, we can only get messages from within our own Hubble volume.) Level 2: If you go far enough, you’ll fall into wonderland In a Level 2 parallel universe, regions of space are continuing to undergo an inflation phase. Because of the continuing inflationary phase in these universes, space between us and the other universes is literally expanding faster than the speed of light — and they are, therefore, completely unreachable. Two possible theories present reasons to believe that Level 2 parallel universes may exist: eternal inflation and ekpyrotic theory. In eternal inflation, recall that the quantum fluctuations in the early universe’s vacuum energy caused bubble universes to be created all over the place, expanding through their inflation stages at different rates. The initial condition of these universes is assumed to be at a maximum energy level, although at least one variant, chaotic inflation, predicts that the initial condition can be chaotically chosen as any energy level, which may have no maximum, and the results will be the same. The findings of eternal inflation mean that when inflation starts, it produces not just one universe, but an infinite number of universes. Right now, the only noninflationary model that carries any kind of weight is the ekpyrotic model, which is so new that it’s still highly speculative. In the ekpyrotic theory picture, if the universe is the region that results when two branes collide, then the branes could actually collide in multiple locations. Consider flapping a sheet up and down rapidly onto the surface of a bed. The sheet doesn’t touch the bed only in one location, but rather touches it in multiple locations. If the sheet were a brane, then each point of collision would create its own universe with its own initial conditions. There’s no reason to expect that branes collide in only one place, so the ekpyrotic theory makes it very probable that there are other universes in other locations, expanding even as you consider this possibility. Level 3: If you stay where you are, you’ll run into yourself A Level 3 parallel universe is a consequence of the many worlds interpretation (MWI) from quantum physics in which every single quantum possibility inherent in the quantum wavefunction becomes a real possibility in some reality. When the average person (especially a science fiction fan) thinks of a “parallel universe,” he’s probably thinking of Level 3 parallel universes. Level 3 parallel universes are different from the others posed because they take place in the same space and time as our own universe, but you still have no way to access them. You have never had and will never have contact with any Level 1 or Level 2 universe (we assume), but you’re continually in contact with Level 3 universes — every moment of your life, every decision you make, is causing a split of your “now” self into an infinite number of future selves, all of which are unaware of each other. Though we talk of the universe “splitting,” this isn’t precisely true. From a mathematical standpoint, there’s only one wavefunction, and it evolves over time. The superpositions of different universes all coexist simultaneously in the same infinite-dimensional Hilbert space. These separate, coexisting universes interfere with each other, yielding the bizarre quantum behaviors. Of the four types of universes, Level 3 parallel universes have the least to do with string theory directly. Level 4: Somewhere over the rainbow, there’s a magical land A Level 4 parallel universe is the strangest place (and most controversial prediction) of all, because it would follow fundamentally different mathematical laws of nature than our universe. In short, any universe that physicists can get to work out on paper would exist, based on the mathematical democracy principle: Any universe that is mathematically possible has equal possibility of actually existing.

Article / Updated 04-27-2023

General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity. Einstein immediately realized that his theory of special relativity worked only when an object moved in a straight line at a constant speed. What about when one of the spaceships accelerated or traveled in a curve? Einstein came to realize the principle of equivalence, and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field. As Einstein later related the discovery, he was sitting in a chair thinking about the problem when he realized that if someone fell from the roof of a house, he wouldn’t feel his own weight. This suddenly gave him an understanding of the equivalence principle. As with most of Einstein’s major insights, he introduced the idea as a thought experiment. If a group of scientists were in an accelerating spaceship and performed a series of experiments, they would get exactly the same results as if sitting still on a planet whose gravity provided that same acceleration, as shown in this figure. Einstein’s brilliance was that after he realized an idea applied to reality, he applied it uniformly to every physics situation he could think of. For example, if a beam of light entered an accelerating spaceship, then the beam would appear to curve slightly, as in the left picture of the following figure. The beam is trying to go straight, but the ship is accelerating, so the path, as viewed inside the ship, would be a curve. By the principle of equivalence, this meant that gravity should also bend light, as shown in the right picture of the figure above. When Einstein first realized this in 1907, he had no way to calculate the effect, other than to predict that it would probably be very small. Ultimately, though, this exact effect would be the one used to give general relativity its strongest support.

Article / Updated 04-14-2023

General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity. The theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of field equations, which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity. As Einstein developed his general theory of relativity, he had to refine the accepted notion of the space-time continuum into a more precise mathematical framework. He also introduced another principle, the principle of covariance. This principle states that the laws of physics must take the same form in all coordinate systems. In other words, all space-time coordinates are treated the same by the laws of physics — in the form of Einstein’s field equations. This is similar to the relativity principle, which states that the laws of physics are the same for all observers moving at constant speeds. In fact, after general relativity was developed, it was clear that the principles of special relativity were a special case. Einstein’s basic principle was that no matter where you are — Toledo, Mount Everest, Jupiter, or the Andromeda galaxy — the same laws apply. This time, though, the laws were the field equations, and your motion could very definitely impact what solutions came out of the field equations. Applying the principle of covariance meant that the space-time coordinates in a gravitational field had to work exactly the same way as the space-time coordinates on a spaceship that was accelerating. If you’re accelerating through empty space (where the space-time field is flat, as in the left picture of this figure), the geometry of space-time would appear to curve. This meant that if there’s an object with mass generating a gravitational field, it had to curve the space-time field as well (as shown in the right picture of the figure). Without matter, space-time is flat (left), but it curves when matter is present (right). In other words, Einstein had succeeded in explaining the Newtonian mystery of where gravity came from! Gravity resulted from massive objects bending space-time geometry itself. Because space-time curved, the objects moving through space would follow the “straightest” path along the curve, which explains the motion of the planets. They follow a curved path around the sun because the sun bends space-time around it. Again, you can think of this by analogy. If you’re flying by plane on Earth, you follow a path that curves around the Earth. In fact, if you take a flat map and draw a straight line between the start and end points of a trip, that would not be the shortest path to follow. The shortest path is actually the one formed by a “great circle” that you’d get if you cut the Earth directly in half, with both points along the outside of the cut. Traveling from New York City to northern Australia involves flying up along southern Canada and Alaska — nowhere close to a straight line on the flat maps we’re used to. Similarly, the planets in the solar system follow the shortest paths — those that require the least amount of energy — and that results in the motion we observe. In 1911, Einstein had done enough work on general relativity to predict how much the light should curve in this situation, which should be visible to astronomers during an eclipse. When he published his complete theory of general relativity in 1915, Einstein had corrected a couple of errors and in 1919, an expedition set out to observe the deflection of light by the sun during an eclipse, in to the west African island of Principe. The expedition leader was British astronomer Arthur Eddington, a strong supporter of Einstein. Eddington returned to England with the pictures he needed, and his calculations showed that the deflection of light precisely matched Einstein’s predictions. General relativity had made a prediction that matched observation. Albert Einstein had successfully created a theory that explained the gravitational forces of the universe and had done so by applying a handful of basic principles. To the degree possible, the work had been confirmed, and most of the physics world agreed with it. Almost overnight, Einstein’s name became world famous. In 1921, Einstein traveled through the United States to a media circus that probably wasn’t matched until the Beatlemania of the 1960s.

Article / Updated 02-07-2023

Many physicists feel that string theory will ultimately be successful at resolving the hierarchy problem of the Standard Model of particle physics. Although it is an astounding success, the Standard Model hasn’t answered every question that physics hands to it. One of the major questions that remains is the hierarchy problem, which seeks an explanation for the diverse values that the Standard Model lets physicists work with. For example, if you count the theoretical Higgs boson (and both types of W bosons), the Standard Model of particle physics has 18 elementary particles. The masses of these particles aren’t predicted by the Standard Model. Physicists had to find these by experiment and plug them into the equations to get everything to work out right. You notice three families of particles among the fermions, which seems like a lot of unnecessary duplication. If we already have an electron, why does nature need to have a muon that’s 200 times as heavy? Why do we have so many types of quarks? Beyond that, when you look at the energy scales associated with the quantum field theories of the Standard Model, as shown in this figure, even more questions may occur to you. Why is there a gap of 16 orders of magnitude (16 zeroes!) between the intensity of the Planck scale energy and the weak scale? At the bottom of this scale is the vacuum energy, which is the energy generated by all the strange quantum behavior in empty space — virtual particles exploding into existence and quantum fields fluctuating wildly due to the uncertainty principle. The hierarchy problem occurs because the fundamental parameters of the Standard Model don’t reveal anything about these scales of energy. Just as physicists have to put the particles and their masses into the theory by hand, so too have they had to construct the energy scales by hand. Fundamental principles of physics don’t tell scientists how to transition smoothly from talking about the weak scale to talking about the Planck scale. Trying to understand the “gap” between the weak scale and the Planck scale is one of the major motivating factors behind trying to search for a quantum gravity theory in general, and string theory in particular. Many physicists would like a single theory that could be applied at all scales, without the need for renormalization (the mathematical process of removing infinities), or at least to understand what properties of nature determine the rules that work for different scales. Others are perfectly happy with renormalization, which has been a major tool of physics for nearly 40 years and works in virtually every problem that physicists run into.

Article / Updated 12-14-2022

For most interpretations, superstring theory requires a large number of extra space dimensions to be mathematically consistent: M-theory requires ten space dimensions. With the introduction of branes as multidimensional objects in string theory, it becomes possible to construct and imagine wildly creative geometries for space that correspond to different possible particles and forces. It’s unclear, at present, whether those extra dimensions exist or are just mathematical artifacts. The reason string theory requires extra dimensions is that trying to eliminate them results in much more complicated mathematical equations. It’s not impossible, but most physicists haven’t pursued these concepts in a great deal of depth, leaving science (perhaps by default) with a theory that requires many extra dimensions. From the time of Descartes, mathematicians have been able to translate between geometric and physical representations. Mathematicians can tackle their equations in virtually any number of dimensions that they choose, even if they can’t visually picture what they’re talking about. One of the tools mathematicians use in exploring higher dimensions is analogy. If you start with a zero-dimensional point and extend it through space, you get a one-dimensional line. If you take that line and extend it into a second dimension, you end up with a square. If you extend a square through a third dimension, you end up with a cube. If you then were to take a cube and extend into a fourth dimension, you’d get a shape called a hypercube. A line has two “corners” but extending it to a square gives four corners, while a cube has eight corners. By continuing to extend this algebraic relationship, a hypercube would be a four-dimensional object with 16 corners, and a similar relationship can be used to create analogous objects in additional dimensions. Such objects are obviously well outside of what our minds can picture. Humans aren’t psychologically wired to be able to picture more than three space dimensions. A handful of mathematicians (and possibly some physicists) have devoted their lives to the study of extra dimensions so fully that they may be able to actually picture a four-dimensional object, such as a hypercube. Most mathematicians can’t (so don’t feel bad if you can’t). Whole fields of mathematics — linear algebra, abstract algebra, topology, knot theory, complex analysis, and others — exist with the sole purpose of trying to take abstract concepts, frequently with large numbers of possible variables, degrees of freedom, or dimensions, and make sense of them. These sorts of mathematical tools are at the heart of string theory. Regardless of the ultimate success or failure of string theory as a physical model of reality, it has motivated mathematics to grow and explore new questions in new ways, and for that alone, it has proved useful.

Cheat Sheet / Updated 06-30-2022

String theory, often called the “theory of everything,” is a relatively young science that includes such unusual concepts as superstrings, branes, and extra dimensions. Scientists are hopeful that string theory will unlock one of the biggest mysteries of the universe, namely how gravity and quantum physics fit together.

Article / Updated 03-26-2016

Although string theory is a young science, it has had many notable achievements. What follows are some landmark events in the history of string theory: 1968: Gabriele Veneziano originally proposes the dual resonance model. 1970: String theory is created when physicists interpret Veneziano’s model as describing a universe of vibrating strings. 1971: Supersymmetry is incorporated, creating superstring theory. 1974: String theories are shown to require extra dimensions. An object similar to the graviton is found in superstring theories. 1984: The first superstring revolution begins when it’s shown that anomalies are absent in superstring theory. 1985: Heterotic string theory is developed. Calabi-Yau manifolds are shown to compactify the extra dimensions. 1995: Edward Witten proposes M-theory as unification of superstring theories, starting the second superstring revolution. Joe Polchinski shows branes are necessarily included in string theory. 1996: String theory is used to analyze black hole thermodynamics, matching earlier predictions from other methods.

Article / Updated 03-26-2016

String theory is a work in progress, so trying to pin down exactly what the science is, or what its fundamental elements are, can be kind of tricky. The key string theory features include: All objects in our universe are composed of vibrating filaments (strings) and membranes (branes) of energy. String theory attempts to reconcile general relativity (gravity) with quantum physics. A new connection (called supersymmetry) exists between two fundamentally different types of particles, bosons and fermions. Several extra (usually unobservable) dimensions to the universe must exist. There are also other possible string theory features, depending on what theories prove to have merit in the future. Possibilities include: A landscape of string theory solutions, allowing for possible parallel universes. The holographic principle, which states how information in a space can relate to information on the surface of that space. The anthropic principle, which states that scientists can use the fact that humanity exists as an explanation for certain physical properties of our universe. Our universe could be “stuck” on a brane, allowing for new interpretations of string theory. Other principles or features, waiting to be discovered.

Article / Updated 03-26-2016

String theory’s concept of supersymmetry is a fancy way of saying that each particle has a related particle called a superpartner. Keeping track of the names of these superpartners can be tricky, so here are the rules in a nutshell. The superpartner of a fermion begins with an “s,” so the superpartner of an “electron” is the “selectron” and the superpartner of the “quark” is the “squark.” The superpartner of a boson ends in “–ino,” so the superpartner of a “photon” is the “photino” and of the “graviton” is the “gravitino.” Use the following table to see some examples of the superpartner names. Some Superpartner Names Standard Particle Superpartner Higgs boson Higgsino Neutrino Sneutrino Lepton Slepton Z boson Zino W boson Wino Gluon Gluino Muon Smuon Top quark Stop squark