I (Physicist) will be at Dragon*Con in Atlanta this year. Rather than a setting up the usual stand, I’ll be wandering around wearing a beat-up, white t-shirt with the words “Ask a Mathematician Ask a Physicist” emblazoned upon the chest in an X.
If you’re there, and happen to see someone fitting that description, feel free to ask a question or just say hi. It’s either a Physicist, which is alright, or else it’s someone with a good sense of humor and a t-shirt printer, which is even better.
Physicist: This question has had a lot of forms, from questions about hot air balloons, to “just hovering in the air”, to weather. But the common thread boils down to “what keeps the atmosphere moving with the surface of the Earth?”.
The short answer is “the ground has drag”, and the slightly longer answer is “sometimes it doesn’t completely”.
First, it’s useful to know what the atmosphere is like (as if you haven’t been breathing it practically all day). It’s a little surprising how much air there isn’t. Although you’ll hear about the atmosphere extending to a hundred miles or more above our heads, it becomes so thin, so fast, that almost none of that “counts”. If all of the atmosphere were as dense as it is at sea level, then it would only be about 7 km tall. People in eight countries could literally walk to space!
Left: the atmosphere (out to 100 km) compared to the Earth. Right: if our atmosphere wasn’t so fluffy and was instead only 7.3 km thick..
The point is that the atmosphere, rather than being a heavenly swath of lung-food, is a tiny puddle of gas, thinly painted on the surface of our world. The Earth for its part is covered in bumps and wrinkles, like mountains, valleys, tress, and whatnot.
The stuff on the surface of the Earth pushes on the wind exactly as hard as the wind pushes on it. So, overall, the air moves with the ground.
These “bumps” catch the atmosphere and keep it moving with the surface. Even if a stationary, non-rotating atmosphere were to suddenly replace ours, it would find itself moving with the rest of the Earth in short order (after the worst storm ever, by far). In physics (reality) there’s no difference between moving and not moving, so a stationary fan is just as good at stopping moving air, as a moving fan is at moving stationary air. Once air is moving with the Earth it’s got momentum, and that’s what keeps it moving (or “what keeps it still”, if you happen to live on Earth).
It turns out that the overwhelming majority of the movement of the atmosphere is tied up in rotating with (and so sitting still relative to) the Earth. The highest wind speed ever verified was 253 mph (that’s gust speed) as measured at Barrow Island. That immediately sounds less impressive when you consider that the wind was measured relative to Barrow Island, which at the time was traveling east at about 940 mph. Still is.
That all said, if you go high enough you find that the surface of the world starts to look pretty smooth. Mountains and seas and whatnot all start to look like the same, fairly smooth surface. As a result, high altitude winds take the turning of the Earth as more of a strong suggestion than as a rule, the way air near the surface does (20 mph gusts! Howsoever shall my hat stay on?). High altitude winds routinely blow at well over 100 mph.
Speaking of which, wind is powered mostly by convection: one region of the world gets warmer, a bubble of hot air rises, nearby air rushes in to take its place, that sort of thing. Wind isn’t caused by the rotation of the Earth, but it is affected by it.
Hurricanes: powered by warm water, pushed into a loop by the Earth.
Everything in space wants to travel in a straight line, so when air from sunny Barrow Island (traveling east at 940 mph) drifts south to also-sunny Perth (traveling east at a mere 850 mph), it finds itself traveling east 90 mph faster than the ground. Usually the difference in eastward speed between two points on the globe gets broken down by the ground, by the time the air has breezed from one to the other. When that east-speed-difference doesn’t get broken down, usually because the air covered the distance too fast, you get a big swirl of air. But keep in mind, ultimately the wind doesn’t get its energy from the Earth, it gets it from heat, which mostly comes from the Sun (“mostly” because warm water and dirt does a lot).
Physicist: Nope!
“Resonance” is a “driven harmonic oscillation“, where the driving force pushes and pulls at, or near, the “resonant frequency” of whatever it is that doing the resonating. There are two big issues involved with destroying stuff using sound, or gentle taps, or whatever you’re using to drive the motion. The first is that nothing in the world “rings” perfectly, and the second is that every example of harmonic motion you’re likely to come across is actually an example of “damped harmonic motion”.
Here’s what “not ringing perfectly” means. You can make anything bounce around and shake, but “harmonic motion” is something very specific. Harmonic motion is a vibration that takes place at just one frequency.
Something that moves harmonically moves back and forth in a sine wave. Maybe more or less spread out or taller than this, but always in exactly this shape.
When something is oscillating back and forth there has to be a “restoring force” to bring it back to center. For a swinging pendulum that force is gravity, for a spring or a wine glass it’s the springiness of the material. The one requirement for harmonic motion is that the restoring force is linear, and proportional to how much the object has been pushed from center. For example, if you pull on a spring by a distance X and it pulls with a force F, then if you pull it twice as far, 2X, it will pull with twice the force, 2F. This is the famous “Hooke’s law“: F = -kX. The negative here means that the force points in the opposite direction of the displacement, so if a pendulum has swung to the right, then gravity is pulling it to the left.
When you change things by just a tiny bit the response is almost always linear. Or at least very nearly linear. If this weren’t the case, then physicists would barely be capable of doing any calculations at all. This is called a “first order approximation” or “linearization”, and it’s really just the statement that things (in a mathematical sense) are smooth.
A typical force response for a spring. If you don’t stretch it too far it reacts linearly (straight line), but if you stretch it too far then it doesn’t (and eventually gets turned into a straight wire or breaks).
Most physical systems have the “linear on a small scale” property. It’s just a question of when it breaks down. This is why clock pendulums don’t swing very far, for example.
So this is the first big problem; if you push something hard enough, or if the oscillation gets too large, then the restoring force won’t be linear. As a result the system starts to lose all of the nice properties that make it a harmonic oscillator. One way for this to happen is for the object to break (huzzah!), but most of the time the oscillation frequency starts getting wonky, the wave stops being pretty (not just one frequency), and trying to induce resonance just sorta stops working.
The second big problem is dampening. Nothing’s perfect so over time every oscillator loses energy. Pluck a guitar string, and it’ll eventually go mute. Stop pushing a kid on a swing, and they’ll eventually just sit there. And maybe get hungry.
Damped harmonic oscillation looks like this.
When you try to make something resonate you’re adding up a bunch of these waves (one for every time you “push the swing”), but because of dampening the waves from earlier don’t add as much as the waves from later. Even if the system is still pretty linear, this puts a cap on how big an oscillation you can get for a given amount of pushing.
So, just to make sure the point on this is too fine, even if the frequency is dialed in perfectly, most things can not be destroyed by resonance.
If the dampening is bad enough, then instead of resonating at all the object will just “ooze” back to where it started. When things are “over-damped” patience completely stops being a virtue and you really need to get all of the energy in place all at once. For example, if you’re using sound you’d need to replace the speaker or your voice with a bomb or a hammer.
The “brown note” is a sound that supposedly resonates intestines and makes a mess. However, there are issues. Entrails don’t ring like bells, they flop like meat. Even if you’re careful to stay in the “linear regime” (very, very small forces and oscillations), you’ll find that meat is usually critically damped, although not always.
There are better ways to induce trou dropping: laxatives, roller-coasters, away-toilet situations, etc.
While it is true that the (American) military has spent some money looking into the brown note (lots of people have), it’s also true that they’ve spent some money looking into almost everything. Statistically speaking, you can’t spend $680,000,000,000 without buying something useless.
Physicist: The weird effects that show up in quantum mechanics (a lot of them anyway) are due to the wave-nature of the world making itself more apparent. What we normally think of as “particle behavior” is just what happens when the waves you’re talking about are very small (compared to what’s around) and are “decoherent” (which means the frequency, phase, and polarization are all pretty random between one photon and the next). It’s a long way from obvious (there’s some math) but, for example, the way light streams through gaps is governed entirely by the wave-nature of light, and not because it’s a particle.
It looks like the sunlight is made of particles moving in straight lines, but in reality you don’t need “particleness” to describe what’s happening here. The gaps in the door are very large compared to the wavelength of the light (approx. 0.01 m vs. 0.0000005 m). With enough elbow room, waves will proceed in a straight line due to interference effects.
Technically, absolutely everything is fundamentally quantum mechanical. What we consider to be “classical mechanics” is just a special case of quantum mechanics (a large-scale, non-coherent case).
But that’s not at the heart of this question.
Light wears its quantum behavior proudly. The De Broglie wavelength decreases with increasing mass, and while even the lightest particles have fantastically small wavelengths (electrons typically have wavelengths on the order of trillionths of a meter), light can have wavelengths ranging up to miles long (radio waves). Observing quantum effects in matter is difficult, but we see it in light so much that we think it’s normal (which it is, I suppose).
So, other than obvious stuff; like chemistry or… absolutely everything, what follows are some distinctly wavy and quantumy things that you might come across while palling around on this big blue world. Also, this list is far more incomplete than complete, so I’m open to suggestions for things to add.
Technology younger than fifty years old (give or take)
There are a lot of well-established, well-known (in the scientific community) quantum mechanical effects that are in use in almost every fancy device devised since the forties. This include things like lasers (Bose-Einstein statistics), tunnel diodes (quantum tunneling), LED lights, and of course exotic stuff like quantum computers. Unfortunately, the qunatum-ness of modern technology is generally pretty well hidden (and thus: boring).
Using cell phones inside (diffusion)
Ever successfully use a cell phone while inside? Radio waves are huge, so their waveness is very apparent. They can ooze around corners (to some extent), and permeate (non-conductive) materials. It’s a little tricky to describe why radios (including cells phones, wifi signals, etc.) work even when there’s no line-of-sight between the sender and receiver, and especially tricky to describe why they work when there’s no path at all (for example, being in a windowless room with the door closed). I wish I could say it’s quantum tunneling (because that would be awesome), but it’s not quite that. It’s similar to the fact that very long, slow ocean waves will cause your boat to rise and fall as a whole, but very short ocean waves are blocked by things like a boat. Radio waves tend to be about the size of buildings and when they come along, rather than just hitting the walls and bouncing off or getting absorbed, they “raise and lower” the electromagnetic field of the whole area. Already it’s practically impossible to describe radio waves in terms of particles.
Pretty colors (thin-film interference)
There’s an optical device called a “Fabry–Pérot interferometer” which uses wave interference to separate out light of very, very nearly equal frequencies, and it’s basically just two mirrors.
As it happens, thin films of transparent material mimic the F-P interferometer.
Some light will bounce off of the surface of the film, and some will go through. If the extra length between paths (red dotted line) is a multiple of the light’s wavelength, then you get “constructive interference”. When you get “destructive interference” then there won’t be light reflected at that angle.
Which of the incoming and outgoing angles experiences constructive interference depends on the thickness of the film, the material of the film, and the color of the light. If the light is “monochromatic” (one color or wavelength, like a laser), then this leads to dark areas. If the light has many different colors, then the areas that are dark for some colors may not be dark for others (which leads to prettiness, see the picture below).
Light can bounce back and forth inside of thin, flat, transparent films. Top Left: a green laser and the glass of a mirror. Top Right: sunlight and an oil film on water. Bottom Left: The path diagram for… Bottom Right: “Newton’s rings”.
It’s sometimes hard to see, but you can see thin-film effects in soap bubbles as well. In that case the thickness of the film tends to change rapidly, which is why the colors in soap films tend to swirl and change so fast.
Soap bubbles: same idea.
True rainbows are formed by another process, but they also ultimately rely on the wave nature of light. It’s just a bit more obscure how.
Dark spots on still lakes (Brewster’s angle and polarized light)
This one is pretty hard to notice. In addition to light’s waviness, it also has polarization (which is a fundamentally not-particle thing to have). The polarization of light affects how it reflects off of a surface (like water) and how it scatters in a gas (like air). If you happen to look at the sky reflecting off of a lake these effects are combined, and at one particular angle they fight each other.
If the angle between the incoming light and the surface of the water is 37°, then only horizontally polarized light will reflect.
The amount of light that reflects off of a surface depends on the polarization of that light, which is why polarized glasses are sold to drivers to cut down on glare. It so happens that if vertically polarized light hits water at about 37° none of it will be reflected (this is called “Brewster’s angle“).
A map of the polarization of the sky (top). The polarization of the blue of the sky “circles the Sun”. So, if you hold up a horizontal polarizer below the Sun it’ll appear clear, but if you hold it up by the side of the Sun it’ll appear dark (bottom).
Because of the way light scatters in air, if you point your hand at any point in the sky (other than the Sun), and turn your palm toward the Sun, then the flat of your hand will be aligned with the polarization of the light coming from that part of the sky. As a result, right around dawn and dusk the entire sky is polarized in the north-south direction.
One consequence of this is that if you’re standing at the right angle early or late in the day, and the sky (not the Sun) is your primary light source, then the face on your digital watch can appear black. Another is that if you look at the sky in a still lake, at about 37° from level, during dawn or dusk, while looking either north or south, you’ll find that the sky isn’t reflected at all and appears black.
For some reason I like this example, in part because it must have been confusing as hell for the occasional fishermen who noticed this over the millennia. However, like the green flash, it’s one of those effects that’s not quite an every-day example.
The door picture is from here, the oil picture is from here, the soap-bubble picture and a description of how it was made can be found here, the reflection picture is from here, and the green-laser-mirror-picture was sent in by a kind-hearted reader as part of a question. The map of the polarizations is from this paper about how bees use that very polarization to navigate. That’s worth pointing out again; many insect species can literally see the polarization of light.
Physicist: The original statement is often something like, “It takes tens of thousands of years for a photon to get from the core to the surface of the Sun, but only eight minutes to get from the Sun to the Earth”. This is one of those great facts that the cognoscenti love to throw around, like “did you know that we only use 10% of our brains?”. Unfortunately, like the 10% thing, there are details behind this fact that make it somewhat less interesting and ultimately either false or not even wrong.
A photon randomly meanders out of the Sun.
The calculation behind the many-thousands-of-years stat goes like this:
-A photon travels, on average, a particular distance, d, before being briefly absorbed and released by an atom, which scatters it in a new random direction.
-Given d and the speed of light, c, you can figure out the average time step and space step size (how often the photon “steps” and how far it “steps” each time).
-The size of the Sun is figured in terms of step size. Some surprisingly tricky math happens, involving “Brownian motion” and probabilities. Finally,
-The average time it would take to get to the surface of the Sun is found.
The math behind this is similar (identical) to the math behind things like Plinko, or the gambler’s ruin. The calculation is a little tricky (which is why it’s sometimes used as an example), but the conclusion is that a photon takes between many thousands and many millions of years to drunkenly wander to the surface of the Sun. If you’re dying of curiosity, then one such calculation is included in the answer gravy (the last part of this post).
However, this result is a little misleading. First, because it makes some subtle mathematical assumptions, and second, because it makes some massive (false) physical assumptions. Inside of the Sun photons are continuously being exchanged, split into many, gathered into one, scattered, and generally not kept in one piece. Rather than thinking of photons in the Sun as being like pinballs bouncing between kicker-like atoms, think of photons as being like over-flowing coffee and the atoms as being like cups.
You can think of the inside of the Sun like this; a bunch of cups overflowing into each other in a giant art thing. It doesn’t make sense to talk about how long it takes for the coffee in the middle-most cup of the art to get out of the art, but it does make sense to talk about how long before half of it is out, or before at least some of it is out.
After being created in a fusion event in the core, the first thing a fresh batch of photons does is get broken up into hundreds of lower energy photons. Talking about how long a photon does anything in the Sun for more than around one nanosecond is a little misleading, because within that time almost every photon in the Sun has been broken apart and/or combined with other photons, leaving them mixed together.
The first tiniest bit of the energy of a photon generated in the core gets to the surface within a couple minutes, and is carried away by photons created there. The many-thousands-of years statistic is useful in that it expresses when about half of a photon’s original energy is bled into space. The last of the photon’s energy is never completely released into space (it’s a “last toothpaste in the tube” sort of thing).
The coffee picture is from here.
Answer Gravy: Just because it’s interesting to see it presented at least one or two ways, at least once, here’s some of the math behind the statistics of random walks and “escapes”. This isn’t the most direct method, but it does the job. Also, photons don’t behave this way, so this is more of a “what if” calculation.
If a particle travels, on average, a distance of Δx in a random direction (50/50 for left/right) every Δt time, then the probability, P(x,t), of the particle being at a particular place at a particular time satisfies the equation:
Now check this out!
This last jump, from a discrete-time process to a continuous process where calculus can be applied, only works when Δx and Δt are very small. In this case the mean free path is Δx = d = 0.01m, which means that at the speed of light, Δt = d/c = 3×10-11s, and 
So, we want to figure out the probability that by some time, T, a particle has passed beyond -R<x<R, where R is the radius of the Sun. Here’s a cute trick: imagine that the Sun keeps going forever, so that nothing special (mathematically difficult) happens at R.
We want to know what the probability is of a photon’s path including x=R (this corresponds to reaching the surface). Notice that if a photon’s path takes it past x=R, then it must have been on the line X=R at some point.
The larger bell curve is the probability of finding the photon at that location after some time t. The pink region is the probability of the photon having gotten to x=R at any time up to T.
Now, since the photon is assumed to have an even chance of going in either direction, then half of the photons that make it to x=R will be on the right of it and half to the left. So, the total probability of making it to x=R is double the probability of being on the far side of x=R. Just so that it looks fancy, let’s say the probability of escape after some time T is 
We can now say that a photon (starting at the center of the Sun, moving at the speed of light, and scattering off of atoms on average every d distance, while hypothetically not being changed by those scatterings) is 50% likely to have escaped the Sun when 
Plugging in 
If you’ve read this far, you may be interested in where there were errors in the big calculation. “Jumping to calculus” introduces a tiny, tiny error, but that decreases rapidly if Δx and Δt are small compared to the scale of the problem (e.g., Δt = 3×10-11 seconds vs. T = 1.1 million years). Also, I’ve ignored the fact that 1) passing -R also means escaping the Sun, and 2) the Sun is a sphere and not a line segment. What I very, very subtly did was “project” the random movement of the photon onto the radial direction, and then only keep track of that direction (“radial” = “out from the center”). This has the effect of changing the 
Overall the errors introduced are smaller than the errors introduced by the range in different quoted values of d (and the fact that it’s assumed to be constant). Still, it involves some cute math.
Physicist: We’ve gotten a handful of questions since this was published and led to articles like this, this, and, this. In a nutshell, some dudes in Germany (Georg Heinze, Christian Hubrich, and Thomas Halfmann) have found a method to shoot a pulse of light, “stop” that light for about a minute and then get the light going again, using “Electromagnetically Induced Transparency” (EIT). It’s not important to know what EIT involves.
So, what does it mean to “stop light”? Light by it’s very nature always travels at the same speed, 2.99 x 108 meters per second: the aptly named “speed of light”. So, for example, when it travels through water or glass and it “slows down” it’s actually just getting absorbed and re-emitted over and over by atoms in whatever it’s traveling through. When it does travel, it travels at full speed. And if “stopping light” means holding the energy for a while and then re-releasing it later, then what’s the difference between what Heinze, Hubrich, and Halfmann did and what a rock does when it heats up in sunlight and then radiates that heat later (as infrared light)?
The answer to that question, and what makes this experiment important, is that the process preserves the photons’ information. The rock that absorbs sunlight and radiates that energy as heat later is scrambling the sunlight’s information completely, but what H and H and H did preserves the light’s information almost perfectly. It’s a little profound how perfectly it preserves the light’s information.
An image, which is “classical information”, is shown after several different storage times. This shows that not only is the energy of the light pulse being stored, but the information in the light pulse is being stored as well.
Not only is it possible to store information like this image, but quantum information can be stored as well. The difference between regular information and quantum information is a little hard to communicate (the exact definition of “information” is already plenty technical before lumping on quantum information). Quantum information is the backbone behind things like entanglement, “action at a distance”, and quantum computation. That last use is the one that physicists are excited about. “Stopping light” is nothing new, but finding a way to store quantum information is a big deal.
Quantum information is the “delicate” part of a thing’s quantum state. When the outside world interacts with a quantum system, it tends to screw it up. We say “the wave function collapsed” or “the system decohered“. Quantum information is only useful and different from classical information when the system (in this case a bunch of light) is allowed to be in superpositions or is entangled with something else. We can tell that the EIT technique preserves quantum information, because we can do experiments on the entanglement between a photon that’s stored, and another that’s not and we find that the entanglement between the two is preserved. Basically, this kind of storage is so “gentle” that it isn’t even a measurement, and all of the quantum state is preserved (quantum information and all).
The UNIVAC I could store as much as 1000 words with 12 letters each in this tank of mercury. This was built right around when “bits” were first becoming the universal standard for information.
Storing light in the way the three H’s have done, is akin to building an early memory device called a “delay line memory“. For comparison, way back in the day (1950 or so) computers were built that used tanks of liquid mercury to store data. Acoustic pulses travel through the mercury tank much slower than electrical signals travel through wires, so you could store tiny chunks of data by reading the pulses from one end of the tank, then re-sending those pulses at the other. The “light storage” technique would be a similar (although much longer time span) memory system for quantum computers.
So, light isn’t being “stopped” it’s “imprinting” on some of the electrons in the crystal that are in very, very carefully prepared states. This imprint isn’t light (so it doesn’t have to move), it’s just excited electrons. That imprint lasts for as much as a minute; slowly accruing errors and fading. After some amount of time that imprint is turned back into light, and it exits the crystal at exactly the speed you’d expect. What makes the experiment most exciting is that this experiment has proven to be an extremely long term method for storing quantum information, which has traditionally been a major hurdle. Normally a quantum computer (such as they are) has to get all of its work done in a fraction of a second.
![\begin{array}{ll} P(x,t) = \frac{1}{2}P(x+\Delta x,t-\Delta t) + \frac{1}{2}P(x-\Delta x,t-\Delta t) \\[2mm] \Rightarrow P(x,t)-P(x,t-\Delta t)=\frac{1}{2}P(x+\Delta x,t-\Delta t)+\frac{1}{2}P(x-\Delta x,t-\Delta t)-P(x,t-\Delta t) \\[2mm] \Rightarrow P(x,t)-P(x,t-\Delta t)=\frac{1}{2}\left(P(x+\Delta x,t-\Delta t)-P(x,t-\Delta t)\right)-\frac{1}{2}\left(P(x,t-\Delta t)-P(x-\Delta x,t-\Delta t)\right) \\[2mm] \rightarrow \frac{\partial P(x,t)}{\partial t}=\frac{\left(\Delta x\right)^2}{2\Delta t}\frac{\partial^2 P(x,t)}{\partial x^2} \\[2mm] \end{array}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++++P%28x%2Ct%29+%3D+%5Cfrac%7B1%7D%7B2%7DP%28x%2B%5CDelta+x%2Ct-%5CDelta+t%29+%2B+%5Cfrac%7B1%7D%7B2%7DP%28x-%5CDelta+x%2Ct-%5CDelta+t%29+%5C%5C%5B2mm%5D++++%5CRightarrow+P%28x%2Ct%29-P%28x%2Ct-%5CDelta+t%29%3D%5Cfrac%7B1%7D%7B2%7DP%28x%2B%5CDelta+x%2Ct-%5CDelta+t%29%2B%5Cfrac%7B1%7D%7B2%7DP%28x-%5CDelta+x%2Ct-%5CDelta+t%29-P%28x%2Ct-%5CDelta+t%29+%5C%5C%5B2mm%5D++++%5CRightarrow+P%28x%2Ct%29-P%28x%2Ct-%5CDelta+t%29%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%28P%28x%2B%5CDelta+x%2Ct-%5CDelta+t%29-P%28x%2Ct-%5CDelta+t%29%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%28P%28x%2Ct-%5CDelta+t%29-P%28x-%5CDelta+x%2Ct-%5CDelta+t%29%5Cright%29+%5C%5C%5B2mm%5D++++%5Crightarrow+%5Cfrac%7B%5Cpartial+P%28x%2Ct%29%7D%7B%5Cpartial+t%7D%3D%5Cfrac%7B%5Cleft%28%5CDelta+x%5Cright%29%5E2%7D%7B2%5CDelta+t%7D%5Cfrac%7B%5Cpartial%5E2+P%28x%2Ct%29%7D%7B%5Cpartial+x%5E2%7D+%5C%5C%5B2mm%5D++++%5Cend%7Barray%7D&bg=ffffff&fg=000&s=0 "\begin{array}{ll} P(x,t) = \frac{1}{2}P(x+\Delta x,t-\Delta t) + \frac{1}{2}P(x-\Delta x,t-\Delta t) \\[2mm] \Rightarrow P(x,t)-P(x,t-\Delta t)=\frac{1}{2}P(x+\Delta x,t-\Delta t)+\frac{1}{2}P(x-\Delta x,t-\Delta t)-P(x,t-\Delta t) \\[2mm] \Rightarrow P(x,t)-P(x,t-\Delta t)=\frac{1}{2}\left(P(x+\Delta x,t-\Delta t)-P(x,t-\Delta t)\right)-\frac{1}{2}\left(P(x,t-\Delta t)-P(x-\Delta x,t-\Delta t)\right) \\[2mm] \rightarrow \frac{\partial P(x,t)}{\partial t}=\frac{\left(\Delta x\right)^2}{2\Delta t}\frac{\partial^2 P(x,t)}{\partial x^2} \\[2mm] \end{array}")
