Should Undergrads Be Exposed To Modern Physics?

A Loyola University undergrad, ca. 1966, studying for a test on special relativity. The class used a text by Hugh D. Young entitled 'Fundamentals of Mechanics and Heat' (which I reviewed at Amazon) and included at least one chapter (14) on special relativity. 

That is perhaps the most that the majority of bright undergrads would be able to cope with even today.


An ongoing controversy in physics education is just how much modern physics today's undergrads should be exposed to. By definition, 'modern physics'  includes all the physics established since 1900, including: quantum theory, special and general relativity and some high energy particle physics (nature of quarks, neutrinos etc.)

The typical plaintive cry, to judge from occasional letters (say to Physics Today)  amounts to some variation on:

"Why must we be stuck in the tedious, boring physics of the 17th, 18th and 19th centuries? Why always Newton, Faraday and such instead of Bohr, Einstein and Schrodinger?  We want to be taught more of the modern physics less of the antiquated stuff!'

Thus, they are chagrined they're still being taught mostly Newtonian mechanics,  and 'old-style' Electricity and magnetism, thermal physics and only a little bit of special relativity.  What they don't say is that even that "little bit" of special relativity often poses problems.

So the question remains: can they really handle more? Most undergrads even at the top universities are simply not at the level they can grasp quantum mechanics or general relativity. Even Richard Feynman ('Feynman Lectures')  while conducting his famous first year course at Caltech, complained about the difficulty of getting quantum concepts over to many of the students.  Of course, there will always be the select few with plenty of mathematics and physics background for whom the content won't present huge problems. But make no mistake they're in the minority.

I still recall teaching a Calculus physics course for which the final chapter of the text assigned had significant quantum mechanics content, including:  the basic Schrodinger wave equation, quantum square well, atomic energy levels, eigenvalues, probability densities and how to compute expectation values. Attached below  is one problem which I worked out for the class:


But which only 1-2 of the class actually got. Most were stung by: a) not having the requisite mathematical ability (which makes one wonder how they managed to be allowed to move on from the earlier semesters) and b) not being able to conceptualize (despite numerous visual aids).


One of the visual aids which flummoxed them is shown below, including the first 3 wave functions for the H- atom(far left), the corresponding probability densities (middle) and the associated energy levels for an "infinite square well".


Basically, the possible energies of the particle, called energy levels, are quantized. The integer n used to designate them, is called the principal quantum number.

The state with lowest energy, n = 1, is called the ground state. The state with n = 2 is called the first excited state and so on.

There are n/2 de-Broglie waves capable of fitting into the well if the particle is in the nth quantum state.

At once the diagram captures the beauty and simplicity of quantum mechanics, in showing how energy is quantized in "jumps" as it is in the actual hydrogen atom.

In retrospect, the experience at Loyola with Hugh Young's presentation  of special relativity was perhaps at the limits of what a well prepared undergrad could understand.   For example, his derivation of the relativistic equation of energy (W) on pages 394, 395:

ò ^x2 _x1  dp/dt  dx = ò ^t2 _t1  (dp/dt)(dx/dt)  dt =

  ò ^t2 _t1  v(dp/dt)   dt  =  ò ^v2 _v1  v(dp/dv)  dv

W=   ò ^v2 _v1   v  d/dv [m_o v/ Ö (1 - v²/c² ] dv

The preceding equation for W applies since for relativistic momentum:

p  =  m_o v/ Ö (1 - v²/c² )

I argue the above is "at the limit" because of the calculus -based approach with which most undergrads would not be comfortable, even those who've taken AP Calculus.  More to the point, if one veers into quantum theory and general relativity, it doesn't get any easier.

This leaves us asking the question of whether there is an alternative solution or approach, specifically for those less able students who aren't attending Caltech, MIT or Harvard.  I believe there is and came away with one idea from a recent Readers Forum article ('How Black Holes Saved My Astronomy Course')  in Physics Today by Joseph Ribaudo..  Ribaudo - based at Utica College, New York- noted the struggle of his students to master the material  with most underperforming (averages in the "C range")  in his astronomy course.   He realized a large part of the problem was that  the existing astronomy textbooks were all too similar in their writing, content and design elements - and often way too mathematical. So he turned to Neil Degrasse Tyson's 'Death By Black Hole' as his central (core) text aided by "reading and writing assignments derived from popular and historical science publications".

Can a similar strategy be used to incorporate more modern physics into undergrad classical physics courses? I am confident this is feasible. Of course, the extent of integration of the modern physics material will hinge on the time allotted for the course or courses.  In most ('General Physics') courses, the modern physics is fit in at the very end - usually in one or two chapters featuring special relativity and some of the 'original' quantum mechanics, say of Bohr then Schrodinger.  At the maximum level of exposure students will be shown the use of the Schrodinger equation to solve for simple QM systems, say a particle in a box. This assumes, of course, the student has done some work in differential equations. (N.B. It is extremely rare for freshman and sophomore physics majors to take a dedicated modern physics course, far less a QM or Relativity course.)

But let's assume students are roughly at the math and abstraction level of  Ribaudo's astronomy class. What texts might supplement the approach to teach modern physics? I'd include the following:

'The Strange Story of the Quantum' - by Banesh Hofman (Dover, 1964)

'Thirty Years That Shook Physics' - by George Gamow (Dover, 1966)

'Relativity and Common Sense' -  by Herman Bondi (Dover, 1964)

'Sidelights on Relativity' - by Albert Einstein (Dover, 1954)

'The ABC of Relativity' - by Bertrand Russell (Signet, 1959)

 All of the above are eminently readable, and the math is at a minimum. Much of the math that is set out, such as in Bondi's book, is in the form of geometry from which simple derivations of the principles are shown. Einstein's book is terrific in providing an understanding at a non-math level by the master himself.

Ribaudo mentions in his article (p. 11)  setting one mid term exam question to the effect: "Discuss the value of reading 'Death By Black Hole' and whether you'd recommend this book to others".  The same sort of question might be asked of any of the books listed above.  In terms of how much contextual mathematics one's students can handle that has to be left to the instructor. For those who might wish to know some QM but minus all the math, Hoffman's book is probably best. For those who might wish to know some special relativity without the math, Russell's book is likely best. For those who are comfortable with plane geometry and some algebra,  Gamow's book is ideal for QM, and Bondi's ditto for special relativity.

If instructors in special relativity wish to get into more problem solving and math exposure - still largely at the intermediate algebra level (with a tiny bit of differential calculus)-  my own text: 'Modern Physics: Notes, Problems and Solutions' is ideal, namely the first three chapters (pp. 1 -45).

Ribaudo also cites "reading and writing assignments" from popular science magazines (e.g. Discover) and this can also provide fertile material for modern physics. For example, asking students to read the excellent article on quantum entanglement in Discover (July/August, 2016, p. 60) and asking students how they might explain this to a friend. Or better, how they might explain it if they could go back in time and encounter Neils Bohr.

In the end, it is possible to present modern physics to undergrads, but it has to be tailored to their own math and abstraction abilities.  If this is done, and there is the time allotted to do it without having to rush, then both physics instructors and their students will be all the better for it. And we may see much less whining from the undergrads about being denied entry into the "modern stuff".

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Addendum:

The best modern physics text I've found for math proficient students - geared to freshman and sophomores - is 'Space, Time and Quanta: An Introduction to Contemporary Physics' by Robert Mills (1994).  In his Introduction,  Mills writes of the text being a "supplement to a more conventional course" or "designed for the physics side of an interdisciplinary course" - i.e. for other (non-physics) students, but don't believe it. The text is in fact perfect as a standalone, totally complete modern physics text - at least on a par with the analogous content in The Feynman Lectures.  The end of chapter problem sets are especially good, and I simply can't see any "interdisciplinary" student being able to get through them.  Hence, I believe Mills greatly underestimates the book's value as a teaching text for modern physics.

College Physics Taught Without Problem Solving? Preposterous!

In the most recent issue of Physics Today (March, p. 10), a 2nd year student in astrophysics at University College,  London  saw his roughly 2-page letter ('How to teach me physics: Tradition is not always a virtue') published. While  acknowledging "physics is the most exciting endeavor I can imagine", Ricardo Heras also wrote:

"The basic courses of my first two years were disappointing. They didn't really give me the opportunity to join that great adventure. Most of my lecturers followed traditional teaching approaches based heavily on solving standard problems and learning by rote, with no hint of free inquiry or discussion. They seemed to be convinced we would understand physics through that method. I was not enthusiastic. "

Mr. Heras then went on to complain that while he and fellow students "spent a lot of time and effort solving textbook-style problems" they didn't really understand physics by doing so and stated that he was "mainly trained to use problem solving techniques."  He then quoted Richard Feynman (of the Feynman lectures fame)  who wrote:

"I don't know what's the matter with people, they don't learn by understanding , they learn by some other way - by rote or something"

Let's first note that the Feynman Lectures in Physics (a  3-volume work) definitely  exemplifies the author’s unconventional approach to physics teaching. But even today most physicists I know look at it as an interesting experiment but only use the texts as  supplemental material to their undergrad courses (whether in QM, Electricity and Magnetism or Thermal Physics) but not as a standalone text.  This is understandable because Feynman drifted all over the place, and didn't follow the usual trajectory for teaching physics, e.g. mechanics, heat, wave motion, optics, electrostatics, E&M, atomic physics, and maybe some quantum physics.

Interestingly, Feynman explored some intriguing problems, such as finding how maser states vary when a maser cavity frequency is nearly - but not exactly- equal to the resonance frequency, w _o (Cf. Vol. III,  Sec. 9.5 ' Transitions off resonance')  Despite numerous such examples scattered over 3 volumes, there was no supplemental problem set or booklet to accompany the lectures. Many have opined they might write such a set eventually, but co-authors Babcock and Leighton never did. .  In any case, it appeared Feynman himself didn't regard having such problems as being of paramount import, as this student Heras doesn't. Indeed, Heras even quotes David Goodstein from a Feb., 1989 Physics Today piece on Feynman:

"If his purpose in giving them was to prepare classes of adolescent boys to solve examination problems in physics, he may not have succeeded particularly well… . If, however, his purpose was to illustrate, by example, how to think and reason about physics, then, by all indications, he was brilliantly successful."


Heras' own frustrations are evident when he writes:

"The aspects of physics I have understood best so far are those I have studied for pleasure. I understood special relativity better when I derived the Lorentz transformations in a different form. This task was much more exciting than the usual assignment of calculating the length contraction of a rod."

But, of course it would be!   The key aspect as well is that when the student studies physics on his own he can apply the creativity and free inquiry he so often finds absent in the class-lecture setting. But this should not be mystifying. Check any university course catalog - even I suspect University College, London - and you will see course listings by credit hours. These give an indication of the time allocated in class for lectures each week. Also labs may be listed separately with their credit hours assigned (Often 1 cr. hr. but the student is actually in lab for 3 hrs.). The whole point is that the university schedule conforms to a specified time frame.  Administrators, for understandable reasons, want to make sure each undergrad - for example - can matriculate in 4-5 years, not take 10 or 15, which Feynman's "create and think it all out" rubric might require.

So it was easy for Feynman to write (as Heras quotes him):

"The best teaching can be done only when there is a direct individual relationship between a student and a good teacher—a situation in which the student discusses the ideas, thinks about the things, and talks about the things."

But let's face it, Feynman is talking about one -on -one tutorials!  Of course that's the best teaching! But how the heck are you going to apply that to a class of say 300 first year calculus physics undergrads - and that's for one section sitting in an auditorium?

Let's also not fool ourselves that problem-solving in physics is not critically important and often discloses how a student is able to think his way through a problem based on using known principles (not "rote").  Conducted with the most interesting, thought absorbing problems, problem solving can be a boon to inculcating physics principles. The object then is not to move away from problems (given they are used at every stage to gauge whether the student can advance - see the end of this post) but to craft better problems!

One such problem  I've given as part of a 2- year General Physics course, is shown below:

A group of 4 astronauts lands on Mars with solar radiation collection material of total area 2000 m ² . If the efficiency of the material is 30%, and the ambient night time temperature on Mars (for their base location at Isidis Planitia) is -40 C (10C day time), will they have adequate collecting material if the solar constant on Mars is 620 W/m² ? (Assume insulating material with a thermal conductivity of 0.08 W/mC, and a need to keep the inside area of their domecile at least at 10 C, requiring solar radiant energy collected of at least 1,200 W per minute for an area of 10 m x 10 m.)

Estimate the thickness of insulating material they're likely to need in order to make it work. Comment on whether this expedition is even feasible given the limits of their materials, and that no more than 100 m ³  of insulating material can be taken.

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The preceding problem clearly makes use of physics principles which the student needs to know to arrive to the solution.  Also, it is clear the student can't just solve the problem by "rote", or by "finding some appropriate equations, putting them together, manipulating them algebraically."  In other words, problem solving need not be mutually exclusive with free inquiry. In fact, a homework problem can afford the opportunity for such inquiry that the limits of course and class structure don't.   Hence, I usually blame physics lecturers for offering uninspired problems for homework and tests, as opposed to creative ones that force the student to go beyond the rote or plug-in paradigm.

While one can sympathize with Heras’ poignant pleas for more “creativity” in physics teaching (especially at the undergrad level), the fact remains that the entire current structure of physics education is founded on mastery of content, as reflected in tests taken at various stages. These determine whether the student is qualified for promotion and even admission to the gateway for ultimate passage (the Ph.D.) which depends on passing a series of comprehensive examination.

To modify this didactic structure in favor of creative in- class learning simply wouldn’t accomplish the goals of physics departments as they are presently structured. For one thing, the time consumed for such learning would surely be much greater  t

han for the current lecture-lab format. Of course, one could assign projects such as I have during my physics teaching career in the 1980s- early 90s, but this is outside of class time. Hence, it does not facilitate learning by supporting independent student creativity in class.

What I have done, to a limited degree, is allow students - such as in general physics, calculus physics or space physics classes I've taught - to design some of their own labs. The design can be presented as a kind of "thought" experiment in the first instance, and then followed up by providing the specific apparatus that would be technically needed to carry it out.

For example, consider the design of an experiment to allow the student to simulate a "subflare", for which I have used the following:

Some solar flare models are based on 'equivalent inductive circuits' in which the circuit is suddenly interrupted or broken when the switch is opened, e.g.


 When the circuit is broken the collapsing flux through the coil tends to maintain the current  I _o hence will generate a spark at the gap if the switch is opened. Suppose the current is rising in simple circuit with a coil, a source of emf, a switch and an inductance, L. Let the current in the simple circuit rise at the rate dI/dt per second. If L is the circuit inductance then the back emf is:

E_b = L (dI/ dt)

The rate at which work is done vs. the back emf is:

 E_b I = LI (dI/ dt)

How might you use this to design an actual  circuit to illustrate how a solar flare occurs via sudden "circuit breaking"? List all the components needed and the specifications.  How would you estimate the magnitude of energy released? How would you re-design the circuit to prevent sparking , i.e. original energy stored in magnetic field of coil now stored in electrostatic field of capacitor? (This latter would be analogous to a double layer in a solar coronal loop which stores excess magnetic free energy. Thus, if its capacitance is large enough  the potential difference across it - and hence across the switch - never rises high enough to cause a spark, or flare in the case of the loop circuit configuration). 

Followup problem: In your lab experiment design, suppose a 1 A current is to be broken (without sparking) in a circuit with self -inductance 1 henry. Find the maximum threshold p.d. across the capacitor, beyond, i.e. which cannot be exceeded. Thereby find the least capacitance that can effectively connected across the switch.

This exercise not only tests the student's creativity and free inquiry skills in simulating a subflare in an inductive circuit, but also how such a flare can be "stifled" under the appropriate physical conditions.  It also addresses Heras' complaint that:

"Traditional teaching methods urge us to perform standard calculations that rarely spark our creativity. Being immersed in such teaching, I feel trapped in a labyrinth whose exit can only be found by solving a ton of mostly uninteresting textbook problems."

Perhaps Heras would have been more at home being challenged in my space physics labs, where free inquiry was given plenty of leeway. In space physics the student is introduced early on to the importance of the Earth's magnetic field, and in particular as the basis for the magnetosphere - on which the aurora depends. In another variant of the earlier experiment, I will set out the following materials  for a space physics lab with no specific instructions for assembly or application: a rectangular coil of at least N = 100 turns mounted on a light wooden frame, a square wooden base 10 cm x 10 cm, two set screws, additional wire, wooden stops, other assorted screws, pins, selected apparatus including flat needle pointer.  If assembled correctly it will appear in finished form below:

The student is then required to construct a working ballistic galvanometer which he or she will then use to find: a) the magnetic flux φ  linking the coil of N turns, the angle of 'dip' and c) the ratio of the vertical component ( B v  ) of Earth's magnetic field to the horizontal component  ( B _H ).   An illustration of the quantities in terms of the coil orientation is shown below, where d  denotes the angle of dip:

The preceding examples are intended to illustrate that there is scope to interject inquiry and creative aspects into labs as well as homework problems. However, I believe it is unrealistic to expect entire classes to be devoted to free inquiry or creative learning via exclusively "first principles" understanding - by which I mean the student is responsible for all or most of the creative effort to learn all his physics "first hand" as it were - with zero outside input.  If we had all the time in the world, or at least more than 4 hours per week for lectures, 3 for labs, plus seven or eight years for physics students to graduate - that might be fine.

But seriously, what physics department today could even remotely entertain such a class or mode of subject delivery?  It would require vastly more time and resources, as well as a totally radical rethinking of physics pedagogy and would come up against the existing system for promotion and qualification, not to mention how we integrate students into the formal university course system. I am not saying it could never work, only that free inquiry and creativity have only limited scope as current physics departments are designed.

Perhaps the optimal time for such exclusive student pursuit of free inquiry is when Heras proceeds on to the pursuit of his Ph.D.  Then, by selecting a problem of inherent appeal, he can develop lines for  original expression of his curiosity, creativity and inquiry not readily available in standard courses. But to expect beginning physics students to do this in more than limited and controlled settings is, frankly, ludicrous.

Of course, to reach that ultimate Ph.D. inquiry point he will have to pass his Ph.D. comprehensive examinations, and that will entail solving a lot of  difficult “traditional” problems! Generally, there will be six exams in six subject areas: classical mechanics, thermodynamics, statistical mechanics, mathematical physics, quantum physics, and electromagnetic theory. These will generally be four hours each.   One example problem from the mechanics exam is shown below:

Even before crossing that threshold, he will need to get through the physics GRE subject exam, which consists of a solid 100 problems to be done in 3 hours. See e.g.

https://www.ets.org/gre/subject/about/content/physics

True, the problems are multiple choice (5 options) but that doesn't mean one can race through them. Consider the example shown below:

The diagram (Fig. 1) shows a resting cylinder of weight W. The coefficient for static friction for all surfaces is 1/3.  The applied force for P = 2W

61)The distance d for which the counterclockwise motion is initiated by P is:

A) d = r/3    B) d = d/2   C) d = r/ 4  D) d = 2d/ 5   E) d = 3r/ 5

62) The vertical reaction force at point A is:

A)    0.3 W  …B)0.5W…..C) 0.8W…. D) 0.9 W…..E) 1.5W

63) The vertical reaction force at point B is:

A) 3W…..B) 2.7 W……C) 1.5 W…..D) 2.1 W….E) 2.5 W

64) The horizontal reaction force at point A is:

A) 1.5 W…..B) 2.1 W….  C) 0.9 W……D) 1.2 W……E) 1.8 W

65) The horizontal reaction force at point B is:

A) 1.5 W…..B) 02.1 W…..C)  0.9 W….D) 1.2 W…..E) 1.8 W

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My point in showing the above? Problems form the core for determining physics advancement at various stages. If you don't like working problems, consider them "too traditional", or whatnot, then physics may not be the "endeavor" for you.

Final note: The Scientific American blog also discussed Heras' letter but framed it in terms of "individualists" vs. "collectivists", i.e. in terms of research bent. Thus, the "individualist" does his own research and publishes his own paper, while the "collectivist" is part of an ensemble of authors - maybe 5 to 15 - who each contribute part of the overall work. But this is the wrong emphasis. No where does Heras even mention research. His complaints are with undergrad physics education and excessive use of traditional problems. His preference is for more "free inquiry" than problem solving, not realizing they need not be mutually exclusive if the problems are designed properly.  As far as research goes, free inquiry (and problem solving) are part and parcel of the process whether one is part of a team ("collective") or on his own ("individualist"). Hence, the SciAm blog mixes apples and oranges in trying to parse Heras' complaints.

Do We Inhabit A Virtual Cosmos? And Can We Prove It?

The question of whether the universe is really an elaborate, multi-dimensional simulation has been around for a number of years.  In a conjecture published some ten years ago[1], Nick Bostrom considered whether the universe and everything within it (including stars, galaxies, planets and human beings.) is one massive, virtual  simulation- a vast and intricate hyper quantum computer program being run from outside- perhaps by an extraordinary intelligence..

Note here that the basis for a quantum computable universe has already been well explicated in books such as: Programming the Universe by  MIT quantum computing engineer Seth Lloyd.  Lloyd estimated the number of 'computer operations' our universe has performed since the Big Bang, every single event that has ever happened, to show how these might emulate 'programming steps'. Of course, as Lloyd has pointed out, this universe simulation can't be replicated. It would require a computer bigger than the universe, though "time would tick more slowly in the program than reality."


In addition, given all entities are mostly empty space anyway,  ultimately predicated on quantum wave forms or wave functions, one can easily see how the ability to manipulate quantum units at will could lead to a simulation of an entire universe. In fact, if one builds consistently on quantum mechanics one can theoretically arrive at all the macroscopic laws of nature that we see govern our universe. For example, in the limit of the quantum number n ® ¥, quantum mechanics converges to Newtonian mechanics, so this rule like others (e.g. Bohr's Complementarity Principle, Heisenberg's Uncertainty Principle, conservation of mass-energy etc.) can easily be built into the simulated system.

  In a sense, if the universe is simulated, then we and all other life forms within it are avatars. And as fanciful as that sounds, some philosophers have long argued that we are more likely to be artificial intelligences, or 3D virtual representations, trapped in a fake universe, than we are to be organically-based minds in a real one. Of course, if these philosophers are correct, then admittedly the concept of a natural afterlife (see previous blog posts from Feb. 9, 10) might be feasible. The reason is that, if the cosmos is a virtual simulation then the very laws of physics (e.g. entropy law) that allow us to devise reality-checking technology and measurements, might have little to do with the rules that govern the meta-cosmos inhabited by our simulators. In other words, if they wish to suspend the 2nd law of thermodynamics to enable a perpetual dream state at death, they can do so.  (But this is the only way I could see that afterlife working!)

But how would one know it or prove it? First things first: Is this equivalence of physical body and cyber-body (avatar) going too far? I don't believe so. The cyber-body is ultimately composed of moving electrons and quanta (photons) that cause multiple pixels on the monitor screen to vary in intensity, producing the illusion of motion. By the same token, the physical body is ultimately constituted of electrons, and can receive and process photons, e.g. by the optic nerve. On a quantum mechanical level the two bodies are very nearly the same. The differences appear at the gross or macroscopic levels, where the physical body displays a 3-dimensional solidity and differentiation of organs, tissues and cells. But are these fundamental? Who can say? It depends on the constructions, doesn't it?

 Assuming clever enough simulations, say using quadrillions of quantum spin system gates operating in tandem, arguably no one would know the difference. In 1982, the Nobel-winning physicist Richard Feynman made a notable (and as it turns out, prescient) observation concerning a “universal quantum simulator”. Feynman conjectured that to obtain 300 nuclear spins, the quantum simulator would need only 300 quantum bits or qubits. So long as one could program the interactions between qubits so that they emulated the interactions between the 300 nuclear spins, the dynamics would be simulated.

Was Feynman off his rocker? Not really! As it turns out one of the best ways to generate simulations via a quantum computer is to use nuclear spins. To see how this could work, study the accompanying diagram with three versions of nuclear spin – to which each is assigned a wave state vector in bracket form. In this case, the nuclear spin down state corresponds to 1>, the spin up state to 0> and there is a combined state: 0 > + 1 >.. (The last is a state of spin along the axis perpendicular to the spin-axis).

More to the point, the latter superposition illustrates the property of quantum parallelism: the ability to compute or register two states simultaneously, which is impossible for normal computers (which register 0 OR 1, never both at one time – unless they are in glitch mode!) Thus, a quantum computer can manipulate two bits (qubits) at one time. If there are a trillion such qubits, each can potentially register 1 and 0 in combined wave bracket states, simultaneously.  It should also be easily seen that this combined wave state is the analog of the superposition of states seen in the electron double slit experiment, e.g. for

                                                   y = y1 + y2

Any qubit in a state (0) can be placed in the double-qubit state by rotating it one fourth of a turn (as evident from inspection of the diagram)  Much more fascinating (and to the point) is that if another qubit is introduced into the scene - say with the same wave state 0> its presence can “flip” the original qubit , effectively producing a quantum-controlled NOT operation which acts like a classical NOT gate. It is billions and billions of such logic gates which form the basis of computing. Given that qubits also hold much more information than ordinary bits, it is easy to see that if such nuclear spins can be used in the sort of logic gate manner described, one can have the basis for a quantum computer.

If one can have such an entity, then one could simulate just about anything. To use the words of Seth Lloyd[2] :

                    

A quantum computer that simulated the universe would have exactly as many qubits as there are in the universe, and the logic operations on those qubits would exactly simulate the dynamics of the universe.


              And further[3]:

Because the behavior of elementary particles can be mapped directly onto the behavior of qubits acting via logic operations, a simulation of the universe on a quantum computer is indistinguishable from the universe itself

This is a profound statement! It implies that if the universe were indeed a mammoth simulation we’d likely never be able to prove it.  But.....what if the simulators, analogous to human beings, aren't 'gods' or perfect? Then it is feasible an imperfect copy of the universe may be running, just good enough to fool most of its inhabitants - at least until they evolved the brain power to conceive applicable tests.

A breakthrough came in 2007, when John D. Barrow, a professor of mathematical sciences at Cambridge University first suggested that an imperfect simulation of reality would contain detectable glitches. Just like this computer I'm now using to blog, the universe's operating system would need updates to keep working. As the simulation degraded, Barrow observed, we might see aspects of nature that are supposed to be constant, i.e. the speed of light, or the fine structure constant, inexplicably 'drift' from their stable values.

Two years ago, Silas Beane and colleagues at the University of Washington, suggested a more concrete test of the simulation hypothesis. Most physicists assume space is smooth and extends out infinitely. But cosmologists modeling the early universe cannot easily re-create a perfectly smooth background to contain their atoms, stars and galaxies.  Instead, they build up their simulated space from a lattice or grid, just as television images are comprised of multiple pixels.

Silas' team calculated that the motion of particles within their simulation, and hence their energy, is related to the distance between the points of the lattice: the smaller the grid size the higher the energy of particles can be. That means that if our universe is a virtual simulation, we will observe a maximum energy for the fastest particles.

As it happens, astronomers have observed that cosmic rays - high speed particles that originate in far flung galaxies- always arrive at Earth's vicinity with a specific maximum energy of about 10²⁰    electron volts.  Recall that:  1.6 x 10^-19 J = 1 eV

So that:  10²⁰ eV  =   10²⁰  eV (1.6 x 10^-19 J /  eV )=       16 J

Another aspect comes to attention: IF space is continuous there is no underlying grid that guides the direction of cosmic rays. In other words, their direction ought to be isotropic - or coming from all directions, equally. Thus, a major test that we live in a simulated cosmos would be a non-isotropic distribution of cosmic rays. It is premature to make any decision here, since astronomers need much more cosmic ray data to make the case one way or the other.

In the meantime, does it make a difference, really - if the cosmos is virtual or real? Not to me. Other than  that I would hope that the virtual simulators, whoever or whatever they are, wouldn't manipulate the 'game' too much!  (I.e. programming in  'demonic' possessions to make certain twits believe demons are real.) Oh, and if they would, I'd rather they pass on letting me have a "natural afterlife"!

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[1] Bostrom:  Philosophical Quarterly.(53),  243 .

[2] Lloyd:  Programming the Universe, 154.
[3] Ibid.