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The quantum double-slit experiment, in which objects are sent toward and through a pair of slits in a wall,and are recorded on a screen behind the slits, clearly shows an interference pattern. It’s natural to ask, “where does the interference occur?”

The problem is that there is a hidden assumption in this way of framing the question — a very natural assumption, based on our experience with waves in water or in sound. In those cases, we can explicitly see (Fig. 1) how interference builds up between the slits and the screen.

Figure 1: How water waves or sound waves interfere after passing through two slits.

But when we dig deep into quantum physics, this way of thinking runs into trouble. Asking “where” is not as straightforward as it seems. In the next post we’ll see why. Today we’ll lay the groundwork.

Independence and Interference

From my long list of examples with and without interference (we saw last time what distinguishes the two classes), let’s pick a superposition whose pre-quantum version is shown in Fig. 2.

Figure 2: A pre-quantum view of a superposition in which particle 1 is moving left OR right, and particle 2 is stationary at x=3.

Here we have

• particle 1 going from left to right, with particle 2 stationary at x=+3, OR
• particle 1 going from right to left, with particle 2 again stationary at x=+3.

In Fig. 3 is what the wave function Ψ(x1,x2) [where x1 is the position of particle 1 and x2 is the position of particle 2] looks like when its absolute-value squared is graphed on the space of possibilities. Both peaks have x2=+3, representing the fact that particle 2 is stationary. They move in opposite directions and pass through each other horizontally as particle 1 moves to the right OR to the left.

Figure 3: The graph of the absolute-value-squared of the wave function for the quantum version of the system in Fig. 2.

This looks remarkably similar to what we would have if particle 2 weren’t there at all! The interference fringes run parallel to the x2 axis, meaning the locations of the interference peaks and valleys depend on x1 but not on x2. In fact, if we measure particle 1, ignoring particle 2, we’ll see the same interference pattern that we see when a single particle is in the superposition of Fig. 1 with particle 2 removed (Fig. 4):

Figure 4a: The square of the absolute value of the wave function for a particle in a superposition of the form shown in Fig. 2 but with the second particle removed. Figure 4b: A closeup of the interference pattern that occurs at the moment when the two peaks in Fig. 4a perfectly overlap. The real and imaginary parts of the wave function are shown in red and blue, while its square is drawn in black.

We can confirm this in a simple way. If we measure the position of particle 1, ignoring particle 2, the probability of finding that particle at a specific position x1 is given by projecting the wave function, shown above as a function of x1 and x2, onto the x1 axis. [More mathematically, this is done by integrating over x2 to leave a function of x1 only.] Sometimes (not always!) this is essentially equivalent to viewing the graph of the wave function from one side, as in Figs. 5-6.

Figure 5: Projecting the wave function of Fig. 3, at the moment of maximum interference, onto the x1 axis. Compare with the black curve in Fig. 4b.

Because the interference ridges in Fig. 3 are parallel to the x2 axis and thus independent of particle 2’s exact position, we do indeed find, when we project onto the x1 axis as in Fig. 5, that the familiar interference pattern of Fig. 4b reappears.

Meanwhile, if at that same moment we measure particle 2’s position, we will find results centered around x2=+3, with no interference, as seen in Fig. 6 where we project the wave function of Fig. 3 onto the x2 axis.

Figure 6: Projecting the wave function of Fig. 3, at the moment of maximum interference, onto the x2 axis. The position of particle 2 is thus close to x2=3, with no interference pattern.

Why is this case so simple, with the one-particle case in Fig. 4 and the two-particle case in Figs. 3 and 5 so closely resembling each other?

The Cause

It has nothing specifically to do with the fact that particle 2 is stationary. Another example I gave had particle 2 stationary in both parts of the superposition, but located in two different places. In Figs. 7a and 7b, the pre-quantum version of that system is shown both in physical space and in the space of possibilities [where I have, for the first time, put stars for the two possibilities onto the same graph.]

Figure 7a: A similar system to that of Fig. 2, drawn in its pre-quantum version in physical space. Figure 7b: Same as Fig. 7a, but drawn in the space of possibilities.

You can see that the two stars’ paths will not intersect, since one remains at x2=+3 and the other remains at x2=-3. Thus there should be no interference — and indeed, none is seen in Fig. 8, where the time evolution of the full quantum wave function is shown. The two peaks miss each other, and so no interference occurs.

Figure 8: The absolute-value-squared of the wave function corresponding to Figs. 7a-7b.

If we project the wave function of Fig. 8 onto the x1 axis at the moment when the two peaks are at x1=0, we see (Fig. 9) a single peak (because the two peaks, with different values of x2, are projected onto each other). No interference fringes are seen.

Figure 9: At the moment when the first particle is near x1=0, the probability of finding particle 1 as a function of x1 shows a featureless peak, with no interference effects.

Instead the resemblance between Figs. 3-5 has to do with the fact that particle 2 is doing exactly the same thing in each part of the superposition. For instance, as in Fig. 10, suppose particle 2 is moving to the left in both possibilities.

Figure 10: A system similar to that of Fig. 2, but with particle 2 (orange) moving to the left in both parts of the superposition.

(In the top possibility, particles 1 and 2 will encounter one another; but we have been assuming for simplicity that they don’t interact, so they can safely pass right through each other.)

The resulting wave function is shown in Fig. 11:

Figure 11: The absolute-value-squared of the wave function corresponding to Fig.10.

The two peaks cross paths when x1=0 and x2=2. The wave function again shows interference at that location, with fringes that are independent of x2. If we project the wave function onto the x1=0 axis, we’ll get exactly the same thing we saw in Fig. 5, even though the behavior of the wave function in x2 is different.

This makes the pattern clear: if, in each part of the superposition, particle 2 behaves identically, then particle 1 will be subject to the same pattern of interference as if particle 2 were absent. Said another way, if the behavior of particle 1 is independent of particle 2 (and vice versa), then any interference effects involving one particle will be as though the other particle wasn’t even there.

Said yet another way, the two particles in Figs. 2 and 10 are uncorrelated, meaning that we can understand what either particle is doing without having to know what the other is doing.

Importantly, the examples studied in the previous post did not have this feature. That’s crucial in understanding why the interference seen at the end of that post wasn’t so simple.

Independence and Factoring

What we are seeing in Figs. 2 and 10 has an analogy in algebra. If we have an algebraic expression such as

• (a c + b c),

in which c is common to both terms, then we can factor it into

• (a+b)c.

The same is true of the kinds of physical processes we’ve been looking at. In Fig. 10 the two particles’ behavior is uncorrelated, so we can “factor” the pre-quantum system as follows.

Figure 12: The “factored” form of the superposition in Fig. 10.

What we see here is that factoring involves an AND, while superposition is an OR: the figure above says that (particle 1 is moving from left to right OR from right to left) AND (particle 2 is moving from right to left, no matter what particle 1 is doing.)

And in the quantum context, if (and only if) two particles’ behaviors are completely uncorrelated, we can literally factor the wave function into a product of two functions, one for each particle:

• Ψ(x1,x2)=Ψ1(x1)Ψ2(x2)

In this specific case of Fig. 12, where the first particle is in a superposition whose parts I’ve labeled A and B, we can write Ψ1(x1) as a sum of two terms:

• Ψ1(x1)=ΨA(x1) + ΨB(x1)

Specifically, ΨA(x1) describes particle 1 moving left to right — giving one peak in Fig. 11 — and ΨB(x1) describes particle 2 moving right to left, giving the other peak.

But this kind of factoring is rare, and not possible in general. None of the examples in the previous post (or of this post, excepting that of its Fig. 5) can be factored. That’s because in these examples, the particles are correlated: the behavior of one depends on the behavior of the other.

Superposition AND Superposition

If the particles are truly uncorrelated, we should be able to put both particles into superpositions of two possibilities. As a pre-quantum system, that would give us (particle 1 in state A OR state B) AND (particle 2 in state C OR state D) in Fig. 13.

Figure 13: The two particles are uncorrelated, and so their behavior can be factored. The first particle is in a superposition of states A and B, the second in a superposition of states C and D.

The corresponding factored wave function, in which (particle 1 moves left to right OR right to left) AND (particle 2 moves left to right OR right to left), can be written as a product of two superpositions:

• Ψ(x1,x2)=Ψ1(x1)Ψ2(x2) = [ΨA(x1)+ΨB(x1)] [ΨC(x2)+ΨD(x2)]

In algebra, we can expand a similar product

• (a+b)(c+d)=ac+ad+bc+bd

giving us four terms. In the same way we can expand the above wave function into four terms

• Ψ(x1,x2)=ΨA(x1)ΨC(x2)+ΨB(x1)ΨC(x2)+ΨA(x1)ΨD(x2)+ΨB(x1)ΨD(x2)

whose pre-quantum version gives us the four possibilities shown in Fig. 14.

Figure 14: The product in Fig. 13 is expanded into its four distinct possibilities.

The wave function therefore has four peaks, one for each term. The wave function behaves as shown in Fig. 15.

Figure 15: The wave function for the system in Fig. 14 shows interference of two pairs of possibilities, first for particle 1 and later for particle 2.

The four peaks interfere in pairs. The top two and the bottom two interfere when particle 1 reaches x1=0, creating fringes that run parallel to the x2 axis and thus are independent of x2. Notice that even though there are two sets of interference fringes when particle 1 reaches x1=0 in all the superpositions, we do not observe this if we only measure particle 1. When we project the wave function onto the x1 axis, the two sets of interference fringes line up, and we see the same single-particle interference pattern that we’ve seen so many times (Figs. 3-5). That’s all because particles 1 and 2 are uncorrelated.

Figure 16: The first instance of interference, seen in two peaks in Fig. 15 is reduced, when projected on to the x1 axis, to the same interference pattern as seen in Figs. 3-5; the measurement of particle 1’s position will show the same interference pattern in each case, because particles 1 and 2 are uncorrelated.

If at the same moment we measure particle 2 ignoring particle 1, we find (Fig. 17) that particle 2 has equal probability of being near x=2.5 or x=-0.5, with no interference effects.

Figure 17: The first instance of interference, seen in two peaks in Fig. 15, shows two peaks with no interference when projected on to the x2 axis. Thus measurements of particle 2’s position show no interference at this moment.

Meanwhile, the left two and the right two peaks in Fig. 15 subsequently interfere when particle 2 reaches x2=1, creating fringes that run parallel to the x1 axis, and thus are independent of x1; these will show up near x=1 in measurements of particle 2’s position. This is shown (Fig. 18) by projecting the wave function at that moment onto the x2 axis.

Figure 18: During the second instance of interference in Fig. 15, the projection of the wave function onto the x2 axis. Locating the Interference?

So far, in all these examples, it seems that we can say where the interference occurs in physical space. For instance, in this last example, it appears that particle 1 shows interference around x=0, and slightly later particle 2 shows interference around x=1.

But if we look back at the end of the last post, we can see that something is off. In the examples considered there, the particles are correlated and the wave function cannot be factored. And in the last example in Fig. 12 of that post, we saw interference patterns whose ridges are parallel neither to the x1 axis nor to the x2 axis. . .an effect that a factored wave function cannot produce. [Fun exercise: prove this last statement.]

As a result, projecting the wave function of that example onto the x1 axis hides the interference pattern, as shown in Fig. 19. The same is true when projecting onto the x2 axis.

Figure 19: Alhough Fig. 12 of the previous post shows an interference pattern, it is hidden when the wave function is projected onto the x1 axis, leaving only a boring bump. The observable consequences are shown in Fig. 13 of that same post.

Consequently, neither measurements of particle 1’s position nor measurements of particle 2’s position can reveal the interference effect. (This is shown, for particle 1, in the previous post’s Fig. 13.) This leaves it unclear where the interference is, or even how to measure it.

But in fact it can be measured, and next time we’ll see how. We’ll also see that in a general superposition, where the two particles are correlated, interference effects often cannot be said to have a location in physical space. And that will lead us to a first glimpse of one of the most shocking lessons of quantum physics.

One More Uncorrelated Example, Just for Fun

To close, I’ll leave you with one more uncorrelated example, merely because it looks cool. In pre-quantum language, the setup is shown in Fig. 20.

Figure 20: Another uncorrelated superposition with four possibilities.

Now all four peaks interfere simultaneously, near (x1,x2)=(1,-1).

Figure 21: The four peaks simultaneously interfere, generating a grid pattern.

The grid pattern in the interference assures that the usual interference effects can be seen for both particles at the same time, with the interference for particle 1 near x1=1 and that for particle 2 near x2=-1. Here are the projections onto the two axes at the moment of maximal interference.

Figure 22a: At the moment of maximum interference, the projection of the wave function onto the x1 axis shows interference near x1=1. Figure 22b: At the moment of maximum interference, the projection of the wave function onto the x2 axis shows interference near x2=-1.

Reader James Blilie has returned with some recent photos of California. James’s captions are indented, and you can enlarge the pictures by clicking on them. The road he traveled down is my favorite one in the U.S., and, I think, the most scenic. I used to travel it when I went from Davis, CA. to Death Valley to collect flies.

Here is a set from our trip to the southern California desert in January 2025.

We again traveled down US 395 through eastern California to the Palm Desert area for some warmth and sunlight to break up the Pacific Northwest winter.  We returned up I-5 through California to Weed, California where we turned off onto US Hwy 97 through eastern Oregon.

These are mostly landscape photos, which is my thing.  As you can tell from the photos, we were lucky with the weather.

Descending to Mono Lake from Conway Summit:

Moonrise over the White Mountains from the Owens River valley, near Bishop, California:

Mount Whitney range from near Lone Pine, California (also in the Owens Valley):

A shot from hiking in the Andreas Canyon, near Palm Springs, California.  The canyons in the San Jacinto range above Palm Springs have flowing rivers and are full of life:

Next are two shots from a hike in Joshua Tree National Park.  Mojave Yucca (Yucca schidigera) and Teddy Bear Cholla (Cylindropuntia bigelovii).  Both of them shouting at you:  “don’t touch!”:

Next are two shots from the Thousand Palms Oasis.  An overview of the site, which has thousands of California Fan Palms (Washingtonia filifera) and then a show of the palm foliage:

Then a few shots from our homeward journey.

At a rest area on northbound I-5 in the Central Valley of California, we found olive trees growing with lots of fallen fruit underneath them.  (Olea europaea):

Mount McLoughlin and Upper Klamath Lake at dawn (Oregon):

Equipment:

Olympus OM-D E-M5 (micro 4/3 camera, crop factor = 2.0)
LUMIX G X Vario, 12-35mm, f/2.8 ASPH.  (24mm-70mm equivalent)
LUMIX 35-100mm  f/2.8 G Vario  (70-200mm equivalent)
LUMIX G VARIO 7-14mm f/4.0 ASPH

The long-standing question of how animals came to have an anus may have been solved by studies of which genes are active during development in various animals

When the James Webb Space Telescope was launched in December 2021, one of its primary purposes was to see the first galaxies in the Universe forming just a few million years after the Big Bang. In true JWST style though, it has surpassed all expectations and now, a team of astronomers think they have gone even further back, seeing one galaxy clearing the early fog that obscured the Universe! The image represents a point in time 330 million years after the Big Bang and reveals a bright hydrogen emission from the fog surrounding a galaxy. It was somewhat unexpected though as current models predict it would have been blown away long ago!

The fashion retailer, H&M, has announced that they will start using AI generated digital twins of models in some of their advertising. This has sparked another round of discussion about the use of AI to replace artists of various kinds.

Regarding the H&M announcement specifically, they said they will use digital twins of models that have already modeled for them, and only with their explicit permission, while the models retain full ownership of their image and brand. They will also be compensated for their use. On social media platforms the use of AI-generated imagery will carry a watermark (often required) indicating that the images are AI-generated.

It seems clear that H&M is dipping their toe into this pool, doing everything they can to address any possible criticism. They will get explicit permission, compensate models, and watermark their ads. But of course, this has not shielded them from criticism. According to the BBC:

American influencer Morgan Riddle called H&M’s move “shameful” in a post on her Instagram stories.

“RIP to all the other jobs on shoot sets that this will take away,” she posted.

This is an interesting topic for discussion, so here’s my two-cents. I am generally not compelled by arguments about losing existing jobs. I know this can come off as callous, as it’s not my job on the line, but there is a bigger issue here. Technological advancement generally leads to “creative destruction” in the marketplace. Obsolete jobs are lost, and new jobs are created. We should not hold back progress in order to preserve obsolete jobs.

Machines have been displacing human laborers for decades, and all along the way we have heard warnings about losing jobs. And yet, each step of the way more jobs were created than lost, productivity increased, and everybody benefited. With AI we are just seeing this phenomenon spread to new industries. Should models and photographers be protected when line workers and laborers were not?

But I get the fact that the pace of creative destruction appears to be accelerating. It’s disruptive – in good and bad ways. I think it’s a legitimate role of government to try to mitigate the downsides of disruption in the marketplace. We saw what happens when industries are hollowed out because of market forces (such as globalization). This can create a great deal of societal ill, and we all ultimately pay the price for this. So it makes sense to try to manage the transition. This can mean providing support for worker retraining, protecting workers from unfair exploitation, protecting the right for collective bargaining, and strategically investing in new industries to replace the old ones. One factory is shutting down, so tax incentives can be used to lure in a replacement.

Regardless of the details – the point is to thoughtfully manage the creative destruction of the marketplace, not to inhibit innovation or slow down progress. Of course, industry titans will endlessly echo that sentiment. But they appear to be interested mostly in protecting their unfettered ability to make more billions. They want to “move fast and break things”, whether that’s the environment, human lives, social networks, or democracy. We need some balance so that the economy works for everyone. History consistently shows that if you don’t do this, the ultimate outcome is always even more disruptive.

Another angle here is if these large language model AIs were unfairly trained on the intellectual property of others. This mostly applies to artists – train an AI on the work of an artist and then displace that artist with AI versions of their own work. In reality it’s more complicated than that, but this is a legitimate concern. You can theoretically train an LLM only on work that is in the public domain, or give artists the option to opt out of having their work used in training. Otherwise the resulting work cannot be used commercially. We are currently wrestling with this issue. But I think ultimately this issue will become obsolete.

Eventually we will have high quality AI production applications that have been scrubbed of any ethically compromised content but still are able to displace the work of many content creators – models, photographers, writers, artists, vocal talent, news casters, actors, etc. We also won’t have to use digital twins, but just images of virtual people who never existed in real life. The production of sound, images, and video will be completely disconnected (if desired) from the physical world. What then?

This is going to happen, whether we want it to or not. The AI genie is out of the bottle. I don’t think we can predict exactly what will happen. There are too many moving parts, and people will react in unpredictable ways. But it will be increasingly disruptive. Partly we will need to wait and see how it plays out. But we cannot just sit on the sideline and wait for it to happen. Along the way we need to consider if there is a role for thoughtful regulation to limit the breaking of things. My real concern is that we don’t have a sufficiently functional and expert political class to adequately deal with this.

The post H&M Will Use Digital Twins first appeared on NeuroLogica Blog.

The outer planets remain somewhat of a mystery and Neptune is no exception. Voyager 2 has been the only probe that has visited the outermost planet but thankfully the James Webb Space Telescope is powerful enough to reveal it in all its glory. With its cameras regularly fixed on Neptune it has even picked up auroral activity in some of its latest images. The data was gathered back in 2023 using Webb’s Near-Infrared spectrograph which detected the tell tale sign of auroral activity, an emission line of trihydrogen cation. The element appears on other giant planets too when aurora are present.

Some researchers propose that countries should start to rack up a carbon debt once they exceed their carbon budget, obliging them to do more to draw down carbon dioxide, but the idea is unlikely to form part of international climate agreements

We Told You So: Vaccines Cause Autism And So Many Other Really Bad Things

The post Science Based Satire: A Sneak Preview Of RFK Jr.’s Vaccine-Autism Study first appeared on Science-Based Medicine.

Can the cosmic rays bombarding the lunar surface be used to identify subsurface water ice deposits? This is what a recent study and iposter presented at the 56th Lunar and Planetary Science Conference (LPSC) hopes to address as a team of researchers developed a novel method called the Cosmic Ray Lunar Sounder (CoRaLS) capable of detecting subsurface lunar water ice deposits that are elusive to current radar systems. This study has the potential to help expand the human presence on the Moon since water ice deposits are currently being focused on the permanently shadowed regions (PSRs) of the Moon for the upcoming Artemis missions.

When researchers asked people to work on a computer with their phones 1.5 metres away, the amount of time they spent on their phone went down – but they just scrolled social media on their laptop instead

What can Venus-like exoplanets, also known as exoVenuses, teach us about our own solar system and potentially finding life beyond Earth, and how can the planned Habitable Worlds Observatory (HWO) provide these insights? This is what a recent study presented at the 56th Lunar and Planetary Science Conference (LPSC) hopes to address as a team of scientists discussed the difficulties of studying exoVenuses and how HWO can help alleviate these challenges by directly imaging them. This study has the potential to help astronomers develop advanced methods for better identifying and understanding potentially life-harboring exoplanets throughout the cosmos.

Way back in 2006, the Spitzer Space Telescope (SST) took an infrared look at a strange object called Herbig-Haro 49/50. It's a jet flowing away from a hot young star. The Spitzer image showed a fuzzy blob at the end of the jet. Was it part of the jet, or something more distant? Recently, the James Webb Space Telescope (JWST) focused its infrared eye on the same object and sent home a fantastic snapshot of this cosmic tornado. It also answered the question about the blob: it turns out to be a distant galaxy, itself bursting with hot young stars.

Mars is cold and dry, but long ago, it was warmer and wetter. Today, its geology is driven by wind and sand, but it was also shaped by water and maybe even glaciers. Glacial activity on Mars was long assumed to be dry, with glaciers frozen right to their beds, scouring the landscape of the Red Planet. But now, researchers think they've found evidence of subglacial melting, where a layer of water forms under the glacier, helping to form various features on Mars.

A team of researchers has developed a groundbreaking high-temperature copper alloy with exceptional thermal stability and mechanical strength. The research team's findings on the new copper alloy introduce a novel bulk Cu-3Ta-0.5Li nanocrystalline alloy that exhibits remarkable resistance to coarsening and creep deformation, even at temperatures near its melting point.

How gravity causes a perfectly spherical ball to roll down an inclined plane is part of elementary school physics canon. But the world is messier than a textbook. Scientists have sought to quantitatively describe the much more complex rolling physics of real-world objects. They have now combined theory, simulations, and experiments to understand what happens when an imperfect, spherical object is placed on an inclined plane.

How gravity causes a perfectly spherical ball to roll down an inclined plane is part of elementary school physics canon. But the world is messier than a textbook. Scientists have sought to quantitatively describe the much more complex rolling physics of real-world objects. They have now combined theory, simulations, and experiments to understand what happens when an imperfect, spherical object is placed on an inclined plane.

Dark matter is a confounding concept that teeters on the leading edges of cosmology and physics. We don't know what it is or how exactly it fits into the Standard Cosmological Model. We only know that its unseen mass is a critical part of the Universe.

Researchers have conducted groundbreaking research on memristor-based brain-computer interfaces (BCIs). This research presents an innovative approach for implementing energy-efficient adaptive neuromorphic decoders in BCIs that can effectively co-evolve with changing brain signals.

Researchers reveal the way our legs adapt to fast movements. When people hop at high speeds, key muscle fibers in the calf shorten rather than lengthen as forces increase, which they call 'negative stiffness.' This counterintuitive process helps the leg become stiffer, allowing for faster motion. The findings could improve training, rehabilitation, and even the design of prosthetic limbs or robotic exoskeletons.

Researchers reveal the way our legs adapt to fast movements. When people hop at high speeds, key muscle fibers in the calf shorten rather than lengthen as forces increase, which they call 'negative stiffness.' This counterintuitive process helps the leg become stiffer, allowing for faster motion. The findings could improve training, rehabilitation, and even the design of prosthetic limbs or robotic exoskeletons.