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In a breakthrough for hydrogen technology, researchers have introduced an innovative electronic fine-tuning approach that enhances the interaction between zinc and ruthenium.

Researchers have tested whether intoxicated people can be reliable witnesses when it comes to identifying a suspect's face after a crime is committed.

The efficiency of wireless charging systems is limited by power loss occurring due to frequency changes in the resonant circuits that enable power transfer. These necessary modulations reduce electromagnetic interference caused by resonant frequencies on other devices. However, conventional strategies for adapting to changing frequencies are inefficient, cost-prohibitive, and impractical. Now, scientists have designed a resonant tuning rectifier that provides a low-cost, efficient solution to stabilize power delivery in wireless power systems.

The efficiency of wireless charging systems is limited by power loss occurring due to frequency changes in the resonant circuits that enable power transfer. These necessary modulations reduce electromagnetic interference caused by resonant frequencies on other devices. However, conventional strategies for adapting to changing frequencies are inefficient, cost-prohibitive, and impractical. Now, scientists have designed a resonant tuning rectifier that provides a low-cost, efficient solution to stabilize power delivery in wireless power systems.

Using AI to produce footage of video games with a consistent world and rules could prove useful to game designers

An analysis of more than 270,000 glaciers worldwide reveals that they have lost around 7 trillion tonnes of ice since 2000, raising sea levels by 2 centimetres

Spreading crushed rocks on fields can absorb CO2 from the air – now chemists have devised a way to turbocharge this process by creating more reactive minerals

Researchers at Microsoft say they have created so-called topological qubits, which would be exceptionally resistant to errors, but their claim has been met with scepticism

The discovery that farming might not have been the catalyst for civilisation means we must completely rethink the timeline of the first complex societies

New research has emerged on the development of a novel membrane distillation system and an adsorbent (a substance that can trap chemicals on its surface) for the removal of hazardous perfluoroalkyl and polyfluoroalkyl substances (PFAS). Scientists utilized carbon-based materials to successfully remove PFAS from water. This innovative approach could contribute to sustainable purification technologies in the future.

AI is reshaping workplaces, particularly in retail. Researchers explored how AI service quality impacts retail employees' innovation, job fit, and satisfaction. Findings show when employees perceive AI as reliable and empathetic, they are more likely to engage in innovative behavior. AI's adaptability also plays a crucial role in enhancing service quality. While reliability strongly supports innovation, transparency and responsiveness had less influence than expected. Empathy in AI systems was found to have a significant positive effect on employee innovation, creating a more engaging work environment. The study underscores AI's potential to drive service innovation in retail.

Ballbots are versatile robotic systems with the ability to move around in all directions. This makes it tricky to control their movement. In a recent study, a team has proposed a novel proportional integral derivative controller that, in combination with radial basis function neural network, robustly controls ballbot motion. This technology is expected to find applications in service robots, assistive robots, and delivery robots.

Ballbots are versatile robotic systems with the ability to move around in all directions. This makes it tricky to control their movement. In a recent study, a team has proposed a novel proportional integral derivative controller that, in combination with radial basis function neural network, robustly controls ballbot motion. This technology is expected to find applications in service robots, assistive robots, and delivery robots.

In less than a year, the American Association of University Professors (AAUP) has completely abandoned its mission, originally conceived

. . . to advance academic freedom and shared governance, to define fundamental professional values and standards for higher education, and to ensure higher education’s contribution to the common good.

Academic freedom is generally understood as the freedom of faculty to research and teach what they want, subject to the requirement that they have to teach their subject and not something else (e.g., no creationism in evolution class), nor engage in irrelevant proselytizing.  “Shared governance” is the making of university rules by groups (usually faculty) rather than constituting rules handed down from on high.  At the University of Chicago, both academic freedom and free speech are connected as two of our several “fundamental principles” installed to ensure an atmosphere of free inquiry without intimidation.

As U. Chicago law professor Tom Ginsburg points out in a Chronicle of Higher Education piece, however, the  AAUP, however, has taken several positions within the last year that are either inimical or orthogonal to academic freedom. To put it frankly, the AAUP has become authoritarian, adhering to “progressive” politics and abandoning those precepts that it once adopted to further academic freedom. The three changes the AAUP has made to this end include these (there are a few other and more minor ones included in the piece):

a.) Abandoned its opposition to academic boycotts

b.) Approved of the use of diversity statements, finding them “compatible with academic freedom”

c.) Averring that institutional neutrality, as embodied in Chicago’s Kalven Report, need not impact academic freedom one way or the other, so one need not adhere to the Kalven principle that the university or parts of it cannot issue ideological, political, or moral statements unless those statements bear directly on the mission of the University.

This is basically abandoning much of what the AAUP was set up to support: academic freedom. These changes are shameful and reprehensible, and all I can guess is that the AAUP, like much of the academic “progressive left,” has gone woke, which means clamping down on freedom of speech and putting in place more authoritarian policies.

You can read the article by clicking on the headline below or find the piece archived here.)

I have written on two of the AAUP’s backsliding before (on boycotts here and on DEI statements here and here), but Ginsburg provides a handy summary of why the AAUP’s actions contradict its mission. In effect, he takes the organization to the woodshed.  I’ll quote him briefly on the three issues discussed above, with Ginsburg’s words indented:

Approving boycotts. (Everyone but the AAUP admits that its tacit approval of boycotts was intended to rubber-stamp the BDS policy of boycotting Israel, including its academics).

The first salvo came last summer, when the committee issued a statement legitimating academic boycotts, reversing a prior position from 2006 that had declared systemic boycotts to be incompatible with academic freedom because they limit the capacity of scholars to collaborate with whomever they choose. That had been a sensible position. But the new iteration of Committee A suggested that academic boycotts were a “legitimate tactic” and were acceptable against colleges that had themselves violated academic freedom. A bitter debate about Israel is the barely veiled subtext. Whatever the proponents of the Boycott, Divest, and Sanctions movement say about it being limited to institutions and not individuals, it has led to hundreds of cancellations of collaborations with and invitations to individual Israeli scholars, both Arab and Jewish, at a time when that country’s democracy is in deep trouble. In other words, the AAUP has endorsed a practice that interferes on the ground with the academic freedom of individual scholars — precisely the outcome the prior committee had foreseen — while claiming to be neutral on the specific issue of Israel.

Okaying diversity statements.

Next, in October, the AAUP blessed diversity statements as compatible with academic freedom. Mandatory diversity statements are in fact orthogonal to academic freedom, as they do not concern research or teaching. Faculty are divided on their use: Some view them as providing mechanisms to enhance racial diversity among the faculty without running afoul of the law, while others see them as devices to ensure ideological homogeneity. There is significant concern about their legality. The AAUP affirmatively defends them: “Meaningful DEI faculty work,” the organization says, “should be evaluated as part of the core faculty duties of teaching, research, and professional service.” It is hard to imagine that any college receiving federal funds will be able to sustain this posture over the next month, much less the next four years. No leader should have to fight for an already controversial enterprise, one essentially unrelated to academic freedom, when the academic enterprise is under existential threat.

I am one of those opposed to diversity statements, as I see them as both compelled speech (a violation of the First Amendment) and as forced adherence to a particular ideology, since DEI has various interpretations that people disagree with.  Mandating their use thus violates both institutional neutrality and free speech, as well as Chicago’s 1972 “Shils Report“, which says that hiring and promotion must be based on criteria of merit: merit in research, merit in teaching, merit in contributing to the intellectual climate of the university, and meritorious service on committees that further the university’s mission or public service that has an intellectual or research component.  At the very least, the AAUP should not have said anything about diversity statements save the second sentence in the paragraph above.  DEI statements should not have been defended as compatible with academic freedom, as that’s a debatable proposition. (See Brian Leiter’s post highlighted at the bottom.)

Allowing violations of institutional neutrality.

Now comes a third statement, this one adopted in January: “On Institutional Neutrality.” Committee A unhelpfully declares that institutional neutrality is “neither a necessary condition for academic freedom nor categorically incompatible with it.” The main feature of its analysis is a rejection of the policies of the University of Chicago. But the statement contains several mischaracterizations, including a grave misunderstanding of academic freedom itself.

As I’ve written manyt times, institutional neutrality as embodied in Chicago’s Kalven report is essential in buttressing free speech, for “official” statements of the university or its parts that aren’t connected with the university’s mission will have the effect of chilling speech. What student, newly hired faculty, or professor up for tenure would take issue with a political or moral view made “official” because it came from the University or one’s department?  As Ginsburg notes:

The university’s 1970 Shils Report defines academic freedom, which is a function of disciplinary expertise, as “the freedom of the individual to investigate, publish, and teach in accordance with his intellectual convictions.” The AAUP, conversely, conceives of academic freedom as allowing collective units within the university to adopt, by majoritarian vote, policies and statements on external issues, so long as those statements are arrived at in the name of “properly shared governance.” As even Committee A [of the AAUP] acknowledges, this practice has the obvious ability to intimidate junior members of the department. It certainly disincentivizes inquiry on the issues in question. It also invites capacious claims of expertise. Nothing, of course, prevents groups of scholars from signing collective statements on anything. But departments are not bearers of academic-freedom rights. When departmental power is deployed to establish orthodoxies, it inevitably disincentivizes dissent and undermines individual inquiry.

There’s more, but you can read the article at the link above.  The upshot is that the AAUP has been ideologically captured and can no longer be counted on to buttress academic freedom. As Ginsburg concludes (bolding is mine):

The Kalven Report warns us that higher education should not become a “second-rate political force.” But the AAUP itself has become a third-rate defender of academic freedom against a powerful enemy. Rather than focusing on the academic freedom of the individual scholar, Committee A emphasizes collective academic freedom, which it conflates with “shared governance.” It offers us a vision of higher education in which departments promiscuously opine on politics, diversity screening is imposed in hiring and promotion, and unlimited encampments have the warrant of academic freedom. Let’s see how that works out. In our moment of crisis, we need principled leaders able to navigate the storm — and to defend real academic freedom.

Greg Mayer just pointed out to me that over at his website Leiter Reports, U of C law professor Brian Leiter agrees completely with Ginsburg. Two short excerpts from his post of October 10, 2024:

Oh how the mighty have fallen; Committee A of the AAUP used to be a reliable defender of academic freedom, but since its capture by the enemies of academic freedom, it has been going downhill fast.   The latest absurd statement in defense of “diversity statements” reflects pretty clearly the influence of UC Davis law professor Brian Soucek (a member of Committee A), whose mistaken views we have discussed many times before (see especially).   Let me quote the appropriately scathing comments of Professor Tyler Harper (Bates College) from Twitter:

The AAUP statement insisting that mandatory DEI statements are compatible with academic freedom—and not political litmus tests—is ridiculous. DEI is not a neutral framework dropped from the sky, it’s an ideology about which reasonable people—including people of color—disagree.  I have benefited from and support affirmative action, and there are some things that fall under the rubric of DEI that I agree with. But pretending that DEI is not a political perspective or framework—when only people of one political persuasion support DEI—is a flagrant lie.  Evaluating a professor’s teaching with respect to their adherence to a DEI framework is a clear violation of academic freedom. DEI is not some bland affirmation that diversity is important and all people deserve accessible education. It’s a specific set of ideas.

Professor Harper adds:  “Recent events should have made clear that professors, particularly those of us on the left, must defend academic freedom without compromise, even when we disagree with how others use that freedom. When academic freedom is softened, we are always the ones who end up losing.”

. . . DEI is an extramural social goal, just as much as being pro-America in MAGA-land is.  Committee A is dead.  We are fortunate that both FIRE and the Academic Freedom Alliance are actually still defending academic freedom.   I would encourage all readers to resign their membership in the AAUP.  It’s a disgrace.

I hasten to add that I am NOT a member! And had I been, I would have resigned the second the AAUP issued the it’s-okay-to-boycott-Israel statement.

Dubbed CADRE, a trio of lunar rovers are set to demonstrate an autonomous exploration capability on the Moon.

An exciting Moon mission launching in the next year will perform a first, deploying multiple rovers. These will talk to each other and a remote base station, demonstrating an autonomous exploration capability.

The three Cooperative Autonomous Distributed Robotic Exploration (CADRE) rovers were recently packaged and shipped from their home at NASA’s Jet Propulsion Laboratory in Pasadena, California. Each about the size of a small suitcase, the CADRE rovers will launch from LC-39A at the Kennedy Space Center in Florida on a SpaceX Falcon-9 rocket with Intuitive Machines’ IM-3 mission in late 2025 or early 2026. The ultimate destination is the enigmatic Reiner Gamma region in the Oceanus Procellarum (Ocean of Storms) region on the lunar nearside.

Robotic lunar rovers go all the way back to the late Soviet Union’s Lunokod-1 rover on the Luna 17 mission in 1970. CADRE, however, will demonstrate that three rovers can work in unison for lunar exploration. This sort of rover network could come in handy, allowing astronaut controllers to one day explore regions too dangerous to venture into.

A CADRE rover undergoes a vibration test ahead of launch. Credit: NASA/JPL A Robotic Lunar Trio

To this end, the Nova-C lander will lower the solar-powered rovers to the surface shortly after touchdown. Engineers equipped each rover with cameras and ground-penetrating radars for exploration. Controllers expect the rovers to last two weeks (14 days) on the surface, from local sunrise to sunset.

“Our small team worked incredibly hard constructing these robots and putting them to the test,” says Coleman Richdale (NASA-JPL) in a recent press release. “We are all genuinely thrilled to be taking this next step in our journey to the Moon, and we can’t wait to see the lunar surface through CADRE’s eyes.”

This will mark Intuitive Machines’ third delivery to the lunar surface. Part of NASA’s CLPS (Commercial Lunar Payload Services) initiative, The company’s IM-1 mission and Nova-C lander Odysseus made an askew landing at the Malapert A crater early last year. The company will make another attempt with the launch of IM-2 next week on February 26th. The mission will carry NASA’s PRIME-1 (Polar Resources and Ice Mining Experiment) with The Regolith and Ice Drill for Exploring New Terrain (TRIDENT) 1-meter drill. The mission is headed to the Shackleton connecting ridge site in the lunar South Pole region.

A Mars rover twin versus a CADRE rover at JPL’s ‘Mars Yard’. Credit: NASA/JPL

Meanwhile, another CLPS mission, Firefly Aerospace’s Blue Ghost will land on the Moon on March 2nd.

The Reiner Gamma landing site is a high priority target for exploration. Astronomers recognize the feature as one of the best known examples of a ‘lunar swirl’. It’s also a known site for localized magnetic anomalies. What causes swirls on the lunar surface isn’t entirely clear. They definitely stand out in stark contrast to the typical pockmarked, cratered surface of the Moon.

The location of the Reiner Gamma landing site on the lunar nearside. Credit: Dave Dickinson (inset: NASA/LRO). What Else is Aboard IM-3?

In addition to CADRE, several other experiments are hitching a rideshare trip to the Moon aboard IM-3. These include Lunar Vertex (LVx), a joint lander-rover also looking to explore the magnetic anomalies of Reiner Gamma, and the Korea Astronomy Space Science Institute (KASI)’s Lunar Space Environment Monitor (LUSEM) which will monitor the near-surface space environment on the Moon. Also on board is a pointing actuator experiment for the European Space Agency’s MoonLIGHT network. This is a precursor to the agency’s Lunar Geophysical Network for laser ranging and pinpoint measurements.

The CADRE Team plus the trio of rovers, headed to the Moon. Credit: NASA/JPL-Caltech

The Moon is about to become a busy place. It’ll be exciting to see CADRE and other missions resume lunar exploration in the coming years.

The post CADRE’s Three Adorable Rovers Are Going to the Moon appeared first on Universe Today.

There will be no readers’ wildlife today, I’m sad to say, as we’ve run our of contributions save those of Robert Lang, and I don’t want to publish the remaining nine every day.  I guess this is a sign that everything is falling apart.  So today, as it’s Wednesday, we have Jesus and Mo:

And today’s Jesus and Mo strip, called “Math 2,” came with the note, “Another 2007 resurrection today, due to other-work overload. Does anyone remomber this one?”

This is a pretty good one, especially “3 X 1 = 1”:

Astronomers have identified sulfur as a potentially crucial indicator in narrowing the search for life on other planets. While sulfur itself is not necessarily an indication of habitability, significant concentrations of sulfur dioxide in a planet’s atmosphere can suggest that the planet is likely uninhabitable, allowing researchers to eliminate it from further consideration.

The discovery of extraterrestrial life remains one of the most sought-after objectives in modern astronomy. However, this is a formidable challenge. The James Webb Space Telescope is unlikely to detect biosignatures—atmospheric gases produced by living organisms—in nearby planets. Similarly, the upcoming Habitable Worlds Observatory will only be able to assess a limited number of potentially habitable exoplanets.

One of the primary obstacles astronomers face is the typically faint nature of biosignature spectra. To address this, they focus on the potential for planets to host life, particularly through the presence of water vapor in their atmospheres. A planet with substantial water vapor may be more likely to support life.

This concept is encapsulated in the “Habitable Zone,” the region around a star where a planet receives just the right amount of radiation: not too little to freeze all water, and not too much to boil it away. In our solar system, Venus lies near the inner edge of the Habitable Zone with surface temperatures exceeding 800 degrees Fahrenheit beneath a dense atmosphere, while Mars resides primarily outside the zone, its water largely trapped in polar ice caps and subsurface reservoirs.

However, detecting water alone poses challenges. For instance, distinguishing between Earth and Venus based solely on atmospheric spectra is difficult due to their similarities when only searching for water vapor.

Recently, a team of astronomers has identified another potentially useful indicator gas for differentiating uninhabitable from possibly habitable worlds: sulfur dioxide. Warm, wet planets like Earth contain minimal sulfur dioxide because it is washed out of the atmosphere by rain. Conversely, Venus also has little detectable sulfur dioxide, as ultraviolet radiation from the Sun converts it into hydrogen sulfide in the upper atmosphere, driving it downwards.

Planets orbiting red dwarf stars present another scenario. These stars emit minimal ultraviolet radiation, allowing sulfur dioxide to persist in the upper atmospheres of dry, uninhabitable planets. Red dwarfs are of particular interest because they are the most common type of star in the galaxy, and many nearby systems, such as Proxima Centauri and TRAPPIST-1, host planets around red dwarfs, making them prime targets for future searches for life.

This new approach involving sulfur dioxide does not identify planets that might harbor life but helps exclude those that likely do not. If significant sulfur dioxide is detected in the atmosphere of a rocky planet orbiting a red dwarf, it suggests a dry, hot world with a thick atmosphere and little to no water, akin to Venus. Such planets can be deprioritized in the search for life.

Conversely, the absence of significant sulfur dioxide may indicate a planet worth further observation for signs of water vapor and potential life.

The quest to find life on other planets will require extensive investigative efforts and unwavering determination. Any method, including the analysis of sulfur dioxide levels to streamline candidate lists, is highly valuable in this endeavor.

The post What’s That Smell? It’s Sulfur – A New Tool For Finding Alien Life appeared first on Universe Today.

Researchers who have been given access to Google's new AI "co-scientist" tool are enthusiastic about its potential, but it isn't yet clear whether it can make truly novel discoveries

Today, the upheavals of plate tectonics continually reshape Earth. When this began is much disputed - and we can’t fully understand how life began to thrive on our planet until we figure it out

What is a wave function in quantum physics?

Such a question generates long and loud debates among philosophers of physics (and more limited debate among most physicists, who tend to prefer to make predictions using wave functions rather than wondering what they are.) I have a foot in both camps, even though I have no real credentials among the former set. But no matter; today I won’t try to answer my own question in any profound way. We can debate the deeper meaning of wave functions another time.

Instead I just want to address the question practically: what is this function for, in what sense does it wave, and how does it sit in the wider context of physics?

Schrödinger’s Picture of the World

Quantum physics was born slowly, in stages, beginning around 1900. The most famous stage is that of 1925, when Heisenberg, along with Born and Jordan, developed one approach, using matrices, and Schrödinger developed another, using a “wave function”. Both methods could predict details of atomic physics and other systems, and Schrödinger soon showed the two approaches were equivalent mathematically. Nevertheless, he (and many others) felt his approach was more intuitive. This is why the wave function approach is emphasized — probably over-emphasized — in many books on quantum physics.

Suppose we want to investigate a physical system, such as a set of interacting subatomic objects that together make up a water molecule. Westart by imagining the system as being in some kind of initial physical state, and then we ask how that state changes over time. In standard first-year undergraduate physics, using the methods of the 17th-19th century, we would view the initial physical state as consisting of the locations and motions of all the objects at some initial time. Armed with that information, we could then calculate precisely what the system would do in the future.

But experimental data on atomic physics revealed that this older method simply doesn’t agree with nature. Some other approach was needed.

In 1920s quantum physics in the style of Schrödinger, the state of the system is (under-)specified by an unfamiliar object: a function on the space of possibilities for the system. This function gives us a complex number for each possibility, whose square tells us the probability for that particular possibility. More precisely, if we measure the system carefully, Schrödinger’s function at the time of the measurement tells us the probability of our measurements giving one outcome versus another.

For instance, suppose the system consists of two particles, and let’s call the possible position of the first particle x1 and that of the second x2. Then Schrödinger’s function will take the form Ψ(x1,x2) — a function giving us a complex number for each of the possible locations of the two particles. (As I’ve emphasized repeatedly, even though we have a system of two particles, there is only one wave function; I’ve given you a couple of examples of what such functions are like here and here.)

If we want to know the probability of finding the first particle at some definite position X1 and the second at a definite position X2 — assuming we do the measurements right now — that probability is proportional to the quantity |Ψ(X1,X2)|2, i.e. the square of the function when the first particle is at X1 and the second is at X2.

If we choose not to make a measurement right away, Schrödinger’s equation tells us how the function changes with time; if the function was initially Ψ(x1,x2; t=0) = Ψ(x1,x2), then after a time T it will have a new form Ψ(x1,x2; t=T) which we can calculate from that equation. If we then measure the positions of the particles, the probabilities for various measurement outcomes will be given by the square of the updated function, |Ψ(x1,x2; t=T)|2.

Schrödinger’s function is usually called a “wave function”. But this comes with a caveat: it’s not always actually a wave…see below. So it is more accurate to call it a “state function.”

Wave Functions Are Not Things

Probably thanks to advanced chemistry classes, in which pictures of atoms are often drawn that suggest that each electron has its own wave function, it is a common error to think that every particle has a wave function, and that wave functions are physical objects that travel through ordinary space and carry energy and momentum from one place to another, much like sound waves and ocean waves do. But this is wrong, in a profound, crucial sense.

If the electrons and atomic nuclei that make up atoms are like characters in a 19th century novel, the wave function is like an omniscient narrator. No matter how many characters appear in the plot, there is only one such narrator. That narrator is not a character in the story. Instead the narrator plays the role of storyteller, with insight into all the characters’ minds and motivations, able to give us many perspectives on what is going on — but with absolutely no ability to change the story by, say, personally entering into a scene and interposing itself between two characters to prevent them from fighting. The narrator exists outside and beyond the story, all-knowing yet powerless.

A wave function describes the objects in a system, giving us information about all the locations, speeds, energies and other properties that they might have, as well as about how they influence one another as they move around in our familiar three-dimensional space. The system’s objects, of which there can be as many as we like, can do interesting things, such as clumping together to form more complex objects such as atoms. As they move around, they can do damage to these clumps; for instance, they can ionize atoms and break apart biological DNA molecules. The system’s wave function, by contrast, does not travel in three-dimensional space and has neither momentum nor energy nor location. It cannot form clumps of objects, nor can it damage them. It is not an object in the way that electrons , photons and neutrinos are objects. Nor is it a field like the electric field, the Higgs field, and the electron field, which exist in three dimensions and whose waves do have momentum, energy, speed, etc. Most important, each system has one, and only one, wave function, no matter how many objects are in the system.

[One might argue that a wave function narrator is less omniscient, thanks to quantum physics, than in a typical novel; but then again, that might depend on the author, no? I leave this to you to debate.]

I wrote the article “Why a Wave Function Can’t Hurt You” to emphasize these crucial points. If you’re still finding this confusing, I encourage you to read that article.

Some Facts About Wave Functions

Here are a few interesting facts about wave functions. I’ll state them mostly without explanation here, though I may go into more details sometime in the future.

• It is widely implied in books and articles that wave functions emerged for the first time in quantum physics — that they were completely absent from pre-quantum physics. But this is not true; wave functions first appeared in the 1830s.
  
  In the “Hamilton-Jacobi” reformulation of Newton’s laws, the evolution of a non-quantum system is described by a wave function (“Hamilton’s characteristic function”) that is a function on the space of possibilities and satisfies a wave equation quite similar to Schrödinger’s equation. However, in contrast to Schrödinger’s function, Hamilton’s function is a real number, not a complex number, at each point in the space of possibilities, and it cannot be interpreted in terms of probabilities. In very simple situations, Hamilton’s function is the argument (or phase) of Schrödinger’s function, but more generally the two functions can be very different.
  
  
• Wave functions are essential in Schrödinger’s approach to quantum physics. But in other approaches, including Heisenberg’s and the later method of Feynman, wave functions and wave equations do not directly appear. (The situation in pre-quantum physics is completely analogous; the wave function of Hamilton appears neither in Newton’s formulation of the laws of motion nor in the reformulation known as the “action principle” of Maupertuis.)
  
  This is an indication that one should be cautious ascribing any fundamental reality to this function, although some serious scientists and philosophers still do so.
  
  
• The relevant space of possibilities of which the wave function is a function is only half as big as you might guess. For instance, in our example of two particles above, even though the function specifies the probabilities for the various possible locations and motions of the objects in the system, it is actually only a function of either the possible locations or the possible motions (more specifically, the particles’ momenta.) If we write it as a function of the possible locations, then the probabilities for the objects’ motions are figured out through a nontrivial mathematical procedure, and vice versa.
  
  The fact that the wave function can only give half the information explicitly, no matter how we write it down, is related to why it is impossible to know objects’ positions and motions precisely at the same time.
  
  
• For objects moving around in a continuous physical space like the space of the room that you are sitting in, waves are a natural phenomenon, and Schrödinger’s function and the equation that governs it are typical of waves. But in many interesting systems, objects do not actually move, and there’s nothing wavy about the function, which is best referred to as a “state function”. As an example, suppose our system consists of two atoms trapped in a crystal, so that they cannot move, but each has a “spin” that can point up or down only. Then
  • the space of possibilities is just the four possible arrangements of the spins: up-up, up-down, down-up, down-down;
  • the wave state function doesn’t look like a wave, and is instead merely a discrete set of four complex numbers, one for each of the four arrangements;
  • the square of the each of these four complex numbers gives us the probabilities for finding the two spins in each of the four possible arrangements;
  • and Schrödinger’s equation for how the state function changes with time is not a wave equation but instead a 4 x 4 matrix equation.

The space of possibilities for two trapped atoms, each with spin that can be up or down, consists only of the above four physical states; Schrödinger’s state function provides a single complex number for each one, and is in no sense wave-like.

• So although the term “wave function” suggests that waves are an intrinsic part of quantum physics, they actually are not. For the design and operation of quantum computers, one often just needs state functions made of a finite set of complex numbers, as in the example I’ve just given you.
  

• Another case where a state function isn’t a wave function in the sense you might imagine is in quantum field theory, widely used both in particle physics and in the study of many materials, such as metals and superconductors. In this context, the state function shows wavelike behavior but not for particle positions, in contrast to 1920s quantum physics. More on this soon.
  
  
• For particle physics, we need relativistic quantum field theory, which incorporates Einstein’s special relativity (with its cosmic speed limit and weird behavior of space and time). But in a theory with special relativity, there’s no unique or universal notion of time. Unfortunately, Schrödinger’s approach requires a wave function defined at an initial moment in time, and his equation tells us how the function changes from the initial time to any later time. This is problematic. Because my definition of time will differ from yours if you are moving relative to me, my form of the wave function will differ from yours, too. This makes the wave function a relative quantity (like speed), not an intrinsic one (like electric charge or rest mass). That means that, as for any relative quantitiy, if we ever want to change perspective from one observer to another, we may have to recalculate the wave function — an unpleasant task if it is complicated.
  
  Despite this, the wave function approach could still be used. But it is far more common for physicists to choose other approaches, such as Feynman’s, which are more directly compatible with Einstein’s relativity.