Post by: Aeris on 29/01/2022 16:24:19
With the introduction out of the way though, let's begin the thread by talking about Quantum Tunneling. Quantum Tunneling you say? You mean the Quantum mechanical phenomenon in which matter and/or energy pop from one location to another because of a fundamental uncertainty in their position? I mean, that's a somewhat simplified way of putting it, but yes, more or less. I honestly don't know that much about this process and just the description of it alone raises quite a few questions in my head. Specifically...
1. So... according to various online sources, Quantum Tunneling can happen at any random time to any collection of particles across short or even huge distances. But like, if that's really the case, why doesn't happen more frequently in the macroscopic world? Like, Quantum Tunneling apparently happens all the time, hundreds and hundreds of time in the Sun during it's nuclear fusion energy generating process, but what about the Sun is so particularly special that it happens THAT much? Are quantum mechanical effects like Quantum Tunneling more likely to take place in high-energy environments like the Sun's core?
2. Let's say, hypothetically speaking, a macroscopic object like a table or a chair undergoes Quantum Tunneling and essentially teleports from one location to another while I'm still looking at it. What would that look like? would the object just pop right out of and then immediately back into existence in the blink of an eye? Would there be some flashy-looking sparkles or a poof of smoke like in movies and TV shows? Would there be some kind of distortion effect that would hurt my eyes?
3. Is the act of something undergoing Quantum Tunneling considered a Thermodynamic Process? Like, again I've been told that this is something that can happen at any given time to basically any arrangement of particles, but even if we were just dealing with a small number of electrons or protons, doesn't the act of them tunneling from one region of space to another in an instant all on their own kind of break several conservation laws? If a macroscopic object disappeared right in front of my eyes and then repapered 20 feet in the air, wouldn't that break the law of conservation of momentum? Is Quantum Tunneling NOT considered to be a Thermodynamic Process? If so, could it theoretically combat the evilness of entropy and save the Universe from heat death (given enough time of course)?
4. Like I said earlier, Quantum Tunneling apparently takes place all the time in the Sun's core during the conversion of Hydrogen to Helium, but like, HOW do we know this? Did we once send a thermally-insulative drone to the Sun that gathered footage of this happening, or is it like Dark Energy where we actually don't know that much about the specifics of this stuff, but do know that something like Quantum Tunneling needs to exist since without something like it, the Sun couldn't physically produce as much energy as it does?
Post by: Halc on 29/01/2022 17:19:32
1. So... according to various online sources, Quantum Tunneling can happen at any random time to any collection of particles across short or even huge distances. But like, if that's really the case, why doesn't happen more frequently in the macroscopic world?The huge distances are improbable, so it is rarely measured when it happens. The short distances happen all the time. All electronics and even your thought processes depend on it.
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Like, Quantum Tunneling apparently happens all the time, hundreds and hundreds of time in the Sun during it's nuclear fusion energy generating process, but what about the Sun is so particularly special that it happens THAT much?It happens all the time everywhere. Perhaps there is a process in the sun that depends on it, so it gets good press there. But in happens in your pencil as well, which doesn't result in a noticeable difference so nobody writes articles about that.
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Are quantum mechanical effects like Quantum Tunneling more likely to take place in high-energy environments like the Sun's core?I don't think it is considered tunneling unless there is some potential barrier that ends up getting crossed. So maybe in the pencil, the same effects are going on (essentially teleporting), but sans barrier, it's just probabilistic brownian motion and not officially considered tunneling. In light of that, it is more likely to take place where such barriers of potential exist to cross. It doesn't seem to go to a place of higher potential, so your chair (below) isn't going to tunnel up onto the table like it does in the poltergeist movie.
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2. Let's say, hypothetically speaking, a macroscopic object like a table or a chair undergoes Quantum Tunneling and essentially teleports from one location to another while I'm still looking at it. What would that look like?Straight out of Douglas Adams books and his improbability drive. It would look exactly like it was teleported: now it's here, now it's there. The distance is insanely improbable even for a small particle. To have every particle in the chair do it that distance, and all at the exact same time is improbable to more zeroes that I can express.
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would the object just pop right out of and then immediately back into existence in the blink of an eye? Would there be some flashy-looking sparkles or a poof of smoke like in movies and TV shows? Would there be some kind of distortion effect that would hurt my eyes?If only the chair did it, there would be a pop when air filled the place where the chair used to be. I know, I actually did the experiment once, with an egg. One wonders what happens to the air and dust that was already at the new location for the chair.
As for the egg, I microwaved a boiled and peeled egg, set the plate on the table, went and got some other stuff, sat down, and touched the egg with a fork. It quantum tunneled away with that characteristic pop of the air filling the vacant space. Unfortunately it didn't all go the same direction. I found a good deal of it on the ceiling of the next room, but none at all on my plate.
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doesn't the act of them tunneling from one region of space to another in an instant all on their own kind of break several conservation laws?It seems to, yes. The chair moving say a cm to the left seems to violate the conservation of the center of gravity of a system with no force acting on it. This is true of even a single particle, so more is going on than just this simplistic view. Tunneling seems to occur always from a higher potential to a lower one, so there's already an external force of sorts making it want to go that way, held in check only by the barrier. So the chair maybe can teleport to downstairs despite the energy barrier (the floor) being in the way. But it isn't going to just move left a cm.
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If a macroscopic object disappeared right in front of my eyes and then repapered 20 feet in the air, wouldn't that break the law of conservation of momentum?Not if still stationary up there. You did break conservation of energy by doing that. My chair going downstairs also does. Where did its potential energy go when it did that? At least it didn't create new energy.
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Is Quantum Tunneling NOT considered to be a Thermodynamic Process? If so, could it theoretically combat the evilness of entropy and save the Universe from heat death (given enough time of course)?Nope. The entropy of the universe goes up oodles of orders of magnitude faster than tunneling events can undo it.
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4. Like I said earlier, Quantum Tunneling apparently takes place all the time in the Sun's core during the conversion of Hydrogen to HeliumThat's a complex process involving many steps. I'm quite aware of two ways to do it (proton-proton reactions and CNO catalyst reactions), but not aware of which of the steps in each method require tunneling.
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HOW do we know this?The quantum gearheads have figured it out. It's apparently impossible by normal means since insurmountable barriers (probably put up by the strong force) cannot be classically crossed.
Post by: Aeris on 29/01/2022 19:57:58
"Nice to see you again. I'm no expert, so forgive my naive and occasionally humorous answers."
Nice to see you again as well. Don't worry about making a little joke or two during your mission to help me. If anything, a little sense of humor helps make learning things easier since it usually shows you really have a passion for what you do.
"The huge distances are improbable, so it is rarely measured when it happens. The short distances happen all the time. All electronics and even your thought processes depend on it."
Wait, is that how radio waves work? I thought it had everything to do with our ability to manipulate the direction of photons from one electronic device to another.
I can't comment much on the bit about thought processes since I don't even know as much about psychology as cosmology, chemistry and physics (hell, I might actually know just a little bit more about Quantum Mechanics than how the brain works and believe me, that is NOT a brag by any means), and I wanna reserve my mind-based questions for the psychology section of this website (or would a question like that fall under Biology?).
"It happens all the time everywhere. Perhaps there is a process in the sun that depends on it, so it gets good press there. But in happens in your pencil as well, which doesn't result in a noticeable difference so nobody writes articles about that."
Aren't pencils made out of billions of molecules and such? If they can undergo Quantum Tunneling, doesn't that mean there's a non-zero chance that they could just someday disappear and reappear right before my very eyes? Yeah, the odds of me winning the lottery 700 times in a row is greater than that happening, but in theory it's possible, right?
"I don't think it is considered tunneling unless there is some potential barrier that ends up getting crossed. So maybe in the pencil, the same effects are going on (essentially teleporting), but sans barrier, it's just probabilistic brownian motion and not officially considered tunneling. In light of that, it is more likely to take place where such barriers of potential exist to cross. It doesn't seem to go to a place of higher potential, so your chair (below) isn't going to tunnel up onto the table like it does in the poltergeist movie."
Fascinating. I wonder though, what exactly constitutes as a potential barrier in this regard?
"Straight out of Douglas Adams books and his improbability drive. It would look exactly like it was teleported: now it's here, now it's there. The distance is insanely improbable even for a small particle. To have every particle in the chair do it that distance, and all at the exact same time is improbable to more zeroes that I can express."
Haven't read that book so I looked up a gif on Google Images and I gotta say, it looks far flashier than I originally thought it was gonna be.
Just sorta curious. HOW improbable is it that my hypothetical scenario would actually take place. Are we talking like, more zeros than terajoules of energy produced by the Sun in an entire year or something (with like a 1 at the end of it?)?
"If only the chair did it, there would be a pop when air filled the place where the chair used to be. I know, I actually did the experiment once, with an egg. One wonders what happens to the air and dust that was already at the new location for the chair.
As for the egg, I microwaved a boiled and peeled egg, set the plate on the table, went and got some other stuff, sat down, and touched the egg with a fork. It quantum tunneled away with that characteristic pop of the air filling the vacant space. Unfortunately it didn't all go the same direction. I found a good deal of it on the ceiling of the next room, but none at all on my plate."
Interesting. Hey just curious, if you had a superpower that involved causing macroscopic objects to Quantum Tunnel away from the area at will, what would be the largest object you could make disappear and then immediately reappear without worrying about death via void filling air currents? Also, did this story actually happen, cause it honestly sounds hilarious.
"it seems to, yes. The chair moving say a cm to the left seems to violate the conservation of the center of gravity of a system with no force acting on it. This is true of even a single particle, so more is going on than just this simplistic view. Tunneling seems to occur always from a higher potential to a lower one, so there's already an external force of sorts making it want to go that way, held in check only by the barrier. So the chair maybe can teleport to downstairs despite the energy barrier (the floor) being in the way. But it isn't going to just move left a cm."
So... even if a macroscopic object hypothetically did Quantum Tunnel in a way that matches my description, it would never do so in a way that violated gravity or the many established conservation laws. Is that right? Did I say that correctly?
"Not if still stationary up there. You did break conservation of energy by doing that. My chair going downstairs also does. Where did its potential energy go when it did that? At least it didn't create new energy."
Actually, you way you worded this made me realize something kinda strange. An object gains more potential energy the higher up and away from gravity it goes, but who or whatever gets that object up from the ground looses energy in the process, so if a macroscopic object DID Quantum Tunnel above the ground, would it just float there forever until something acts on it again?
"Nope. The entropy of the universe goes up oodles of orders of magnitude faster than tunneling events can undo it."
So... Quantum Tunneling IS theoretically capable of violating the second law of thermodynamics?
"The quantum gearheads have figured it out. It's apparently impossible by normal means since insurmountable barriers (probably put up by the strong force) cannot be classically crossed."
The Strong Force? As in, the Strong Nuclear Force that confines quarks into hadron particles like protons and neutrons and then binds them together to create atomic nuclei? It has the ability to create potential barriers as well? I did not know that.
Thanks so much for answering btw. Really do appreciate it :)
Post by: Halc on 29/01/2022 23:16:06
Wait, is that how radio waves work?No. Radio waves is light moving at c, not a particle tunneling through a barrier, same as say light getting to us from another star.
Transistors use quantum tunneling, as do nerve receptors, required for brain function, not that it alone explains how a brain works, but nerves and transistors are not entirely dissimilar.
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Aren't pencils made out of billions of molecules and such? If they can undergo Quantum Tunneling, doesn't that mean there's a non-zero chance that they could just someday disappear and reappear right before my very eyes? Yeah, the odds of me winning the lottery 700 times in a row is greater than that happening, but in theory it's possible, right?Same as the chair, yes. It's got to go somewhere, but not necessarily all to the same place. Maybe the pencil will just reverse direction, but not the words printed on it. Maybe only the words change. Imagine a pencil that, by pure chance, gives the answers to the test you're taking with it, but reverts to "Bob's pencil #2" whenever the prof gets suspicious and tries to figure out what you're doing. It says unkind things about him as soon as he turns his back. In fact, it says the unkind things all the time, but only on the side away from him.
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Fascinating. I wonder though, what exactly constitutes as a potential barrier in this regard?Usually an electromagnetic potential greater than the energy of the (typically) electron. Remember that circuits can be modeled with water and pipes and such? Such a barrier is like the walls of the canal which are higher than the pressure of the water can cross, but nevertheless it gets wet outside the wall. That's tunneling.
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"Straight out of Douglas Adams books and his improbability drive."It's from Hitchhiker's Guide to the Galaxy. The device generated an improbability field which was first used to at parties to simultaneously teleport all the particles of the undergarments of the hostess half a meter to the left, which helped get the party going. It was later pressed into service more practically as a spaceship drive.
Haven't read that book so I looked up a gif on Google Images and I gotta say, it looks far flashier than I originally thought it was gonna be.
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Just sorta curious. HOW improbable is it that my hypothetical scenario would actually take place. Are we talking like, more zeros than terajoules of energy produced by the Sun in an entire year or something (with like a 1 at the end of it?)?Terajoules is a paltry 12 zeros of joules. I have no idea. Calculate the probability of the particle moving a distance X, and then the probability of the direction of that distance being exactly Y and at exactly time T. You've already got an awful lot of zeros. Now take the number of particles in your chair and raise it to the power of the big number you got just above.
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Hey just curious, if you had a superpower that involved causing macroscopic objects to Quantum Tunnel away from the area at will, what would be the largest object you could make disappear and then immediately reappear without worrying about death via void filling air currents? Also, did this story actually happen, cause it honestly sounds hilarious.If I had a superpower, the limit of the size and consequences of exercising the power would be exactly the amount needed to make the plot interesting. Hence the Enterprise running out of fuel only when the plot demanded it.
As for the egg, yes, that really happened. I was really impressed that the egg could hold that much energy in check.
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So... even if a macroscopic object hypothetically did Quantum Tunnel in a way that matches my description, it would never do so in a way that violated gravity or the many established conservation laws. Is that right? Did I say that correctly?You said it fine, and I don't know the rules. A single particle 'teleporting' seems to violate certain conservation laws, so one wonders if there has to be a counter-reaction somewhere to compensate. Sure, the chair moves a meter to the left, but also the trash can empties itself all over the kitchen floor to the right. Sigh...
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An object gains more potential energy the higher up and away from gravity it goes, but who or whatever gets that object up from the ground looses energy in the process, so if a macroscopic object DID Quantum Tunnel above the ground, would it just float there forever until something acts on it again?It falls if not sitting on something. If it gained classical potential energy by moving up a few meters, one really has to figure out where that energy came from. I don't know, and if it comes from nowhere, then the scenario is probably more than just insanely improbable.
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So... Quantum Tunneling IS theoretically capable of violating the second law of thermodynamics?Any process is. If I kick a pile of blocks, there's a chance they all land in a nice neat Parthenon. The kick isn't even necessary. It's all just really improbable. The 2nd law is all about statistics, which can always be locally violated.
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The Strong Force? As in, the Strong Nuclear Force that confines quarks into hadron particles like protons and neutrons and then binds them together to create atomic nuclei? It has the ability to create potential barriers as well? I did not know that.All the forces can create potential barriers. Gravitational potential barrier keeps the water in my mug. I don't know where tunneling comes into play with the generation of Helium in the sun, so I'm guessing as to what the barriers are.
Post by: Eternal Student on 30/01/2022 03:46:04
Fascinating. I wonder though, what exactly constitutes as a potential barrier in this regard?Just about anything. Quantum mechanics describes a situation where there is a wave function, usually written Ψ and a "potential" usually written V. The potential is just a function of position and time. If you like mathematics V = V(x,t). That's all there is to it. The potential is what will create your barrier.
As Halc stated, the simplest systems to model involve something like a charged particle in an electric field, so V(x,t) is just the electrical potential at position x and time t. However the potential can be anything you like, it could be a hybrid of a potential due to the strong nuclear force plus some potential due to some electric field potential... whatever you're modelling at the time. I've not seen gravitational potential being used but I don't see why it couldn't be. The source of the potential hardly matters, it's just "the potential" which exists throughout space and time and the particle would experience this potential if it was there (and at that time).
If we take your chair example and simplify the situation so that we are only asking the electrons in it to tunnell to somewhere else, then the empty space just to the left of the chair is "the barrier". The chair has some electrons bound to its atoms. Assuming the nucleii don't move, then we can find some approximation for the potential those electrons experience, most of it will be an elctrostatic potential. The electrons have a certain energy and will usually be found in certain places where the potential is low, this will be around the atoms in the usual electron clouds. (The exact solution is going to be impossible to calculate but very roughly we should get a solution like the sort of thing chemists draw for electron orbitals in molecules). That potential has some value everywhere in space (and time) and indeed there is some small chance of finding a few electrons well away from their usual orbital cloud positions. The usually empty space just to the left of the chair is a region of such high potential that a typical electron should rarely be found there. Under classical (not qunatum) mechanics the electron should never be there because it would need an energy that is greater than it actually has (under classical mechanics the electron has effectively escaped from the atom which would demand that it has been given the full ionisation energy normally due for that atom and that electron starting from it's ususal orbital). Anyway, the point is that the empty space just left of the chair is "the barrier", it's the region where the potential exceeds the energy, E, that is nominally available for the electron(s).
Now, let's go back to considering a simpler system where there is only one particle and some potential V(x,t). The system has a certain energy, E, and in classical mechanics there would be no way that the particle you are modelling could be found or should exist in any place or time where the potential exceeds the Energy, E, that was available. It simply does not have the energy to get there or exist there. In Quantum mechanics things are a bit different: The particle that is being modelled with the wave function, ψ(x,t), can sometimes be found in a region where the potential exceeds the nominal amount of Energy, E, that was available. Any region where the potential exceeds the energy, E, is called a "barrier". In classical mechanics it would be very much a hard barrier, there's no way the particle can get beyond it or through it. In Quantum mechanics a barrier has less effect on the wave function. Instead of forcing the wave function all the way down to absolute 0 (which is saying "there's no chance of finding the particle here"), the wave function will usually just fall off rapidly as you move into the barrier, typically you see an exponential fall off with distance and the wave function is often said to become "evanescent".
An ideal or infinite barrier is one where the potential becomes infinite in value and it's a discontinuous jump (it's finite all the way for x < barrier location and then precisely at x = barrier location the potential jumps straight up to ∞ and stays there for x> barrier location). There's probably no such barrier in the real world but it's a useful idea for the mathematics. For an ideal or infinite barrier, the wave function is forced to absolute 0 value the moment you enter the barrier. Only these ideal or infinite barriers act like true hard barriers in Quantum Mechanics. Anyway, there's not thought to be any real life situation where a potential field shows behaviour like this, the potential may become very large and over a short amount of distance but it doesn't ever go vertical (making a discontinuous jump over 0 distance) and it never reaches an infinite value, just a large but finite value. So... as far as quantum mechanics is concerned that's not a hard barrier, the particle can be found inisde that region (with a probability that falls off exponentialy as you progress further into the barrier).
Quantum tunnelling is not just about finding a particle inside a region that should be a barrier. Usually you have a situation where there are two regions of fairly low potential. Two patches of space where the particle could easily exist since the system has Energy, E, greater than that potential. If the barrier between these two regions is only finite (non ideal) then the evanescent wave function (recall that it's falling off exponentially with distance) will reach the other low potential region. Once it get's there, it's fine it doesn't decay anymore. The solution for the wave equation should give you perfectly satisfactory stationary states etc. This is the sort of thing that happens when a particle is created in the first region and fired toward a barrier but there is a perfectly low enough potential region on the other side of the barrier. What you get on the other side of the barrier is a perfectly stable wave function but just with much lower amplitude then the incident wave function that was sent into the barrier (most of the wave function is simply reflected back into the original patch - but some gets through).
Wiki has a few good diagrams and animations for this:
(https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/E14-V20-B1.gif/330px-E14-V20-B1.gif)
(https://upload.wikimedia.org/wikipedia/commons/5/50/EffetTunnel.gif)
The more general situation that is considered to be Quantum tunneling doesn't involve "firing" a localised quantum packet at a barrier. Instead you just have a situation where there is a particle with energy, E, and you place no other restrictions on it (don't try to fire it at the barrier, just leave it to exist wherever it can and solve the wave equations to see what you get). The potential will represent a situation where there are two patches of space with a reasonably low potential and some region between them which is the barrier. This barrier can even be an ideal or infinte barrier. Anyway, the solutions you get indicate that the wave function will be non-zero (just to clarify it has time dependance so it might be 0 some of the time at a given place but it's not 0 all the time in that place) in both regions of low potential. This indicates that there is some chance of finding the particle in either region. If the barrier was ideal (an infinite square wall barrier) then the chances of finding the particle anywhere between those two regions is always 0, yet somehow the particle can be found in either of the two regions. You could find it in the first region today but the second region tomorrow etc. It's NEVER going to be found in between the two, just in one or the other. In this respect if we try to explain this as if the wave was a physical particle then we can imagine that the particle tunnels between the two regions in some amazing way, it seems capable of disappearing from one and re-appearing in the other. (However, it's never actually going to do that while you are observing the particle, under observation it stays in region 1 or region 2, it isn't going to relocate until the system is left unobserved. This is a seperate issue, I hardly know why I mentioned it except that everyone always wants to "see" the thing disappear and re-appear - but this thing is more like a magician doing something behind the curtain, you can't see it happening but when it's done the doves have moved from their original cage to other cage).
That's about all I know about Quantum mechanics and tunnelling. I can put some equations down but I don't think anyone will want to see that. Anyway they're in books and they explain it better than I can.
I can already see that Wikipedia has sections about types of behaviour that they consider to be "Qunatum tunnelling" that I know nothing about.
This is a long post, I'm signing off for now.
Best Wishes.
Post by: evan_au on 30/01/2022 04:26:26
Quote from: OP
hundreds and hundreds of time in the SunThis possibly refers to the reaction proton + proton = deuterium + positron + neutrino (+ energy)
- At the extreme temperatures and density in the center of the Sun, protons run into each other at high speed all the time, but the positive charge of the protons keeps them apart. In empty space, the electrostatic force has infinite range, but in the dense proton/electron soup in the core of the Sun, it has an effective range of the distance to the nearest pair of electrons. This is a potential barrier to protons colliding.
- The strong nuclear force is attractive on very small distances, but repulsive at slightly longer distances (and falls to zero quickly). This is another barrier to two protons colliding.
- Very rarely (the rate-limiting step), one of the protons can decay in that short instant when they are in close proximity, and form deuterium, which is stable. This decay is governed by the weak nuclear force, which has an extremely short range. This is another barrier to the formation of Deuterium.
There is energy released in this process (seen as kinetic energy), so it obeys conservation of energy
- Momentum is balanced (in fact it was "missing" momentum that led to the discovery of the neutrino)
- Some of that energy is carried off to distant parts of the cosmos by the neutrino, the positron annihilates in a pair of gamma rays, while the kinetic energy of the deuterium creates heat, the lowest form of energy. So entropy increases.
https://en.wikipedia.org/wiki/Stellar_nucleosynthesis#Hydrogen_fusion
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electronics...how radio waves work?Most modern semiconductors are based on Field-Effect Transistors (FETs), which have an electrostatic barrier produced by the Gate terminal, blocking electrons or holes from traveling between Source and Drain terminals.
- However, a small number of electrons do tunnel through this electrostatic barrier.
- So by small changes in the Gate voltage, you can produce big changes in the current flowing between Source and Drain, producing power amplification.
- In your cellphone, many stages of amplification produce the transmitted signal, and also amplify and decode the received signals.
- In fact, this tunneling, or "leakage current" is a real barrier to progress in semiconductors, as increasing the density of chips requires making smaller FETs. Because they are physically smaller, tunneling becomes more likely, so they consume more power, even when they are supposed to be "OFF".
- So a small amount of tunneling allows transistors to work, but too much tunneling means that they no longer work.
https://en.wikipedia.org/wiki/Field-effect_transistor
The radio waves themselves don't have a barrier to overcome. The sea of electrons move freely with in the metal antenna; oscillating voltages from the transmitter circuitry push these electrons to & fro. This oscillating current and voltage creates disturbances in the electromagnetic field that propagate off "to infinity".
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pencilsPencils contain graphite (carbon) and wood (which also contains carbon).
- the wood contains a small fraction of carbon 14, which is (very slightly) radioactive, and decays with a half-life of 5,730 years
- It is energetically favourable for the C14 to decay, but the Strong Nuclear Force creates a potential barrier.
- Every second, some C14 atoms in the pencil wood will overcome this barrier, and decay
- All the C14 in the graphite decayed long ago.
- These are single-atom quantum tunneling, by distances around the size of a nucleus.
- So the whole pencil won't move
- And it especially won't move by any width that you are capable of observing.
https://en.wikipedia.org/wiki/Radiocarbon_dating
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what exactly constitutes as a potential barrier?It can be any force - examples above have been an electrostatic barrier, the strong nuclear force or weak nuclear force.
Gravity might provide a potential barrier in the vicinity of a black hole (eg Hawking radiation), but in our solar system, gravity potentials are so weak that we never see it as a barrier to be overcome - normal atomic vibrations occur up-and-down just as well as side-to-side.
As the energy of objects increases, they can overcome increasingly strong barriers:
- At room temperature, atoms in air can travel upwards at 1000km/h, allowing you to hear sounds below you. But near absolute zero, the atoms move much more slowly, and have more trouble overcoming the gravitational potential barrier.
- But SpaceX can overcome Earth's gravitational potential barrier - with immense effort. And New Horizons overcame the Sun's gravitational potential barrier.
- At room temperature, electrons in a FET can overcome a potential barrier amounting to Megavolts per meter (over a distance of nanometers). But if you try to operate them at too high a temperature, the thermal energy of the electrons is routinely enough to overcome the potential barrier, and you get more current, increasing the temperature further= thermal runaway.
- In the center of the Sun (or an atomic bomb), Protons can overcome the strong nuclear force.
- Atoms of Uranium routinely overcome the potential barrier of the Strong Nuclear force, and it fissions into smaller atoms.
- Neutrinos can (very rarely) overcome the (very short-range) potential barrier of the Weak Nuclear force, and interact with normal matter.
Oops! Overlap with 2 posts!
Post by: Aeris on 30/01/2022 12:55:01
"Usually an electromagnetic potential greater than the energy of the (typically) electron. Remember that circuits can be modeled with water and pipes and such? Such a barrier is like the walls of the canal which are higher than the pressure of the water can cross, but nevertheless it gets wet outside the wall. That's tunneling."
So it's kinda like the whole "Heat only moves from hot to cold" thing? The origin point of whatever is about to Quantum Tunnel needs to be in a state of higher energy than the end point? Is this right?
"It's from Hitchhiker's Guide to the Galaxy. The device generated an improbability field which was first used to at parties to simultaneously teleport all the particles of the undergarments of the hostess half a meter to the left, which helped get the party going. It was later pressed into service more practically as a spaceship drive."
Gonna go on a limb here and say that such a device is either straight up impossible in the real world or if it is possible, it will be a lot less impressive than what you're describing. Also, if the device was used in the story to teleport the undergarments of the victim a smidge to the left, would the wearers move as well?
"Terajoules is a paltry 12 zeros of joules. I have no idea. Calculate the probability of the particle moving a distance X, and then the probability of the direction of that distance being exactly Y and at exactly time T. You've already got an awful lot of zeros. Now take the number of particles in your chair and raise it to the power of the big number you got just above."
I'm far to jaded from college homework to do any serious mathematics atm. The point still stands though, right? It's unimaginably unlikely to actually happen in our world, but still not quite impossible.
"If I had a superpower, the limit of the size and consequences of exercising the power would be exactly the amount needed to make the plot interesting. Hence the Enterprise running out of fuel only when the plot demanded it.
As for the egg, yes, that really happened. I was really impressed that the egg could hold that much energy in check."
Yeah, I'll admit, even this hypothetical scenario is just a little bit too unrealistic to take answer seriously, but as someone who love watching, reading and writing hard science-fiction/fantasy stories with physics-abiding magic systems and plausible forms of technology, I am genuinely curious as to how potential dangerous an ability like this would actually be in real-life. Like, could Quantum Tunneling the entirety of Mount Everest create a world-destroying, pressurized blast wave? Could I stand right next to a house undergoing Quantum Tunneling and not worry about accidently killing myself? Again, batsh** insane as all hell, but that's kinda why I love it so much.
"It falls if not sitting on something. If it gained classical potential energy by moving up a few meters, one really has to figure out where that energy came from. I don't know, and if it comes from nowhere, then the scenario is probably more than just insanely improbable."
The improbable in the second sentence implies that not only could something like this actually happen, but that if it did happen, it would in fact be in direct violation of the law of conservation energy and the law of conservation of momentum. At least, that's how I interpreted it.
"Any process is. If I kick a pile of blocks, there's a chance they all land in a nice neat Parthenon. The kick isn't even necessary. It's all just really improbable. The 2nd law is all about statistics, which can always be locally violated."
The way you describe entropy here makes it seem like the second law of thermodynamics has everything to do with the orderliness of the universe. I always thought it had everything to do with the availability of useful energy in the universe. Or are both of these descriptions ultimately mean more or less the same thing in this regard?
Post by: Eternal Student on 31/01/2022 05:49:15
So it's kinda like the whole "Heat only moves from hot to cold" thing? The origin point of whatever is about to Quantum Tunnel needs to be in a state of higher energy than the end point? Is this right?Well I'm not Halc but I'll offer this answer:
No, that's not an absolute requirement.
However, it is a statistical law and in that respect, yes.
See the typical example illustrated and animated in Wikipedia (https://en.wikipedia.org/wiki/Quantum_tunnelling), that's where the animated diagram came from that was put into an earlier post.
A wave packet incident on a finite barrier has a non-zero transmission coefficient and the easiest case to analyse is where the potential of regions I and III are exactly the same (see later diagram).
Some of the information in Wikipedia is a bit scruffy at the moment and there's an entire paragraph that is just adrift and incomplete with the usual warnings "citation needed".
It would be best to get a back-up reference. Here's one from my book-shelf:
- - - - - - - - - - - - - -
Extract from Section 3.3, starting on page 59 of "Quantum Mechanics" by Franz Schwabl, publisher Springer-Verlag, 1990.
.... We now investigate motion in the prescence of the square potential barrier...
V(x) = V0 . Θ(a - |x|) where Θ(z) = 0 if z<0, Θ(z) = 1 if z>0.
(https://oer.physics.manchester.ac.uk/QM/Notes/jsmath/square_barrier.png)
[Image taken from oer.physics.manchester.ac.uk - because I couldn't be bothered to re-create Schwabl's diagram with paintbrush but this one is showing the same information]
A classical particle would be reflected from the barrier for E<V0. In contrast according to quantum theory, we even find a finite transmission probability in this case... This purely quantum mechanical phenomena is known as the tunelling effect.
[page 62, QM, Franz Schwabl]
- - - - - - - - - -
These transmission probabilities (or coefficients) are determined using a continuity requirement for the wave function Ψ and it's derivative Ψ' from which a system of equations is obtained that must be simultaneously satisified. This is usually written in matrix form. There's really no need to put that mathematics in this post but you can read about it in plenty of other places if you wish. It's sufficient to note that you can get some transmission when the potentials are different and there's no requirement that the potential in region III is less than the potential in region I.
However (because there's always a however...) the wave function will tend to have a larger amplitude in the region with the lower potential. What this means is that, all things being equal, a particle left to exist anywhere it can will be found in the region of low potential more often. So, let's imagine this in terms of a statistical average, for a large number of particles thrown into a potential like this (but let's add an infinite step potential at the far left and far right to make sure the particles have to continue bouncing around the central region(s) and not just travelling off to infinity) then there will be a net movement to the region of lower potential from the region of higher potential (until an equilibrium is reached with more particles in the region of lower potential than in the region of higher potential).
So it's kinda like the whole "Heat only moves from hot to cold" thing?Oddly enough, yes it is very much like that. Since that law of thermodynamics can be regarded as just a statistical law anyway.
I'm far to jaded from college homework to do any serious mathematics atm. The point still stands though, right? It's unimaginably unlikely to actually happen in our world, but still not quite impossible.Yes. That is the general idea.
Already too long, signing off for now. Best Wishes.
Post by: Colin2B on 31/01/2022 09:58:50
So it's kinda like the whole "Heat only moves from hot to cold" thing? The origin point of whatever is about to Quantum Tunnel needs to be in a state of higher energy than the end point? Is this right?Sorry, I haven’t had time to read through all the answers you’ve been given, but this is wrong. The energy of the particle is the same on both sides of the barrier, it doesn’t move from higher energy to lower energy. What is different is the wave function which defines the probability of the particle being in a particular location.
This diagram shows the difference:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html
This diagram uses the step function potential barrier which is not correct in real life, but is easier to model mathematically. In reality you get something like this:
Barrier tunnel 300.png (29.78 kB . 300x246 - viewed 6216 times)The origin is the nucleus and electrons are between it and the barrier. The wave function will define the probability of the particle being at position x, so take a very simple probability distribution, I've shown 3 in red at different energy levels (again very, very simplified):
As you can see, at low energies the probability that the electron will be outside the barrier is unlikely, but at higher levels the barrier is thinner and more likely to get a particle outside the barrier.
Also, the lower the mass of the particle the greater the probability it can tunnel (wider spread of the probability distribution.
So for tunnelling:
Thin barrier = greater probability (depends on material)
Higher energy = greater probability
Lower mass = greater probability.
By the time you get to macro sized objects eg pencil and a barrier like a wall, the probability has dropped to so many zeros that it won’t happen. You could throw a pencil at a wall once a second for a billion years and it won’t go through. Of course, many of these maths functions never actually go to zero, but don’t hold your breath.
Also, if the device was used in the story to teleport the undergarments of the victim a smidge to the left, would the wearers move as well?You need to impart energy to the object, the ‘victim’ is heavier than the undergarments, but even flimsy lace panties would be too heavy to teleport. I find the best way to impart the necessary energy is a smooth downward motion, I recommend you try it.
I'm far to jaded from college homework to do any serious mathematics atm. The point still stands though, right? It's unimaginably unlikely to actually happen in our world, but still not quite impossible.So unimaginably low that it is impossible - subject to comment above!.
The best way to teleport a pencil from A to B is to pick it up & throw it, works every time ;D
Unless there is a wall in the way ???
Post by: hamdani yusuf on 31/01/2022 13:17:29
Quote
https://en.wikipedia.org/wiki/Quantum_tunnelling#The_tunneling_problemIf the barrier energy is increased to 40, will there be a portion of the wave packet passes through the barrier?
(https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/E14-V20-B1.gif/330px-E14-V20-B1.gif)
A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.
Post by: Eternal Student on 31/01/2022 13:55:31
This is going to get difficult for Aeris (or anyone else) to make sense of. I'll spend a moment to try and pull the comments from Halc, Clin2B and myself together a bit.
Several of us picked this quote to represent something Aeris asked:
So it's kinda like the whole "Heat only moves from hot to cold" thing? The origin point of whatever is about to Quantum Tunnel needs to be in a state of higher energy than the end point? Is this right?However, Halc was always talking about the potential when he first discussed the situation. In classical mechanics (not qunatum mechanics) the potential energy is only one contribution to the total energy of the particle.
Eternal Student was also careful to talk about the potential.
Aeris was the only one to say "Energy" but most of us knew that they were talking only about the potential.
Colin2B has spoken about the (total) Energy, E, of the particle.
Here's the overall summary:
Halc suggested the tunneling would occurr from region I (where the particle started from) to region III (where the particle ends up) only if the potential in region I was higher than region III.
Eternal Student (E.S. or "me") disagreed with that.
E.S. also suggested that Halc's comment would be reasonable on a statistical basis (so Halc doesn't have to feel too bad about suggesting it in the first place).
Colin2B hasn't commented on that directly but once again has shown a diagram and hyperlink where the potentials in the two regions are exactly the same.
Colin2B has commented that the (total) Energy, E, of the particle remains constant on both sides of the barrier.
E.S. agrees with that and would actually strengthen it: The Energy remains constant wherever the particle is found. Even if you found it inside the barrier it would still hold.
(This is going to take a moment to explain since a classical mechanics view of things would make it impossible for the potential V that the particle is experiencing at some place and time to be greater than the total energy of the system. It would look like the kinetic energy had to be some negative value in order for the total Energy, E, to be less than V. This might be a topic for later discussion since Aeris has already asked about conservation laws like the conservation of energy but for now I'm going to walk away and leave it there. The Energy, E, remains constant).
Best Wishes.
Post by: Eternal Student on 31/01/2022 14:04:59
If the barrier energy is increased to 40, will there be a portion of the wave packet passes through the barrier?Yes. (In line with the previous post, it's the barriers potential you are talking about and not the energy of the particle). The transmitted wave packet will have a smaller amplitude.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html
Is actually a very useful link. (It was found by Colin2B and used in his earlier post). It offers an online calculator to determine transmission coefficients when a particle approaches a barrier of given height and width.
The transmission coefficient is approximately the ratio of (the square of) the amplitude of the transmitted wave to the (square of the) amplitude of the incident wave. This decreases as the potential barrier height increases.
Best Wishes.
Post by: hamdani yusuf on 01/02/2022 03:16:53
Yes. (In line with the previous post, it's the barriers potential you are talking about and not the energy of the particle). The transmitted wave packet will have a smaller amplitude.
Quote
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Fquantum%2Fimgqua%2Fbarr.gif&hash=873722aea2fe8cdd792375f41ab00b04)I think we can only compare two parameters if they have the same dimension. You can't say that 1 Volt is bigger than 1 Joule meaningfully.
According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden.
If you state it as barrier potential, it's unclear what the dimension is.
Post by: Eternal Student on 01/02/2022 12:29:16
If you state it as barrier potential, it's unclear what the dimension is.Sadly this terminology is used in the literature. The "potential" is shown on the diagram in units of energy (for example Joules not Volts). So it is the potential energy that the particle would have if it was physically located there.
It's clear what you meant by the barrier energy but having just received the long post from Colin2B it would be wise to avoid saying anything that looks like E and carries the name "Energy" since that is usually reserved for the Energy of the particle. The Energy of the particle is just called "the Energy", it almost always has the symbol E and it would be the total energy = potential energy + kinetic energy under classical mechanics.
Best Wishes.
Post by: Colin2B on 01/02/2022 12:43:16
This is going to get difficult for Aeris (or anyone else) to make sense of.Unfortunately the nature of discussion vs structured textbook or lecture, unpicking a thread can be hard work.
Aeris was the only one to say "Energy" but most of us knew that they were talking only about the potential.Thanks for confirming that Aeris had misunderstood. I didn't have time to read the whole thread as I'm working on new course at the moment. My concern was that Aeris seemed to be thinking that energy was lost or that it was necessary for the particle to 'roll downhill'
Colin2B has spoken about the (total) Energy, E, of the particle.
Colin2B has commented that the (total) Energy, E, of the particle remains constant on both sides of the barrier.Agreed. Problem is Aeris is dipping in rather than taking the trouble follow up and understand what has been said and he lacks any real background.
E.S. agrees with that and would actually strengthen it: The Energy remains constant wherever the particle is found. Even if you found it inside the barrier it would still hold.
I can see a number of areas where he might get the wrong idea. Take the image linked by hamdani, if Aeris doesn't understand what the wavepacket represents it would be easy to assume this diagram shows only part of the energy passing through the barrier.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.htmlYes, very useful. I regularly recommend it to my students, with the proviso that they work out longhand first and use this to check.
Is actually a very useful link. (It was found by Colin2B and used in his earlier post). It offers an online calculator to determine transmission coefficients when a particle approaches a barrier of given height and width.
Yes, use same units, but fortunately the calculator contains a conversion function so enter whichever is convenient for you.
I think we can only compare two parameters if they have the same dimension. You can't say that 1 Volt is bigger than 1 Joule meaningfully.
If you state it as barrier potential, it's unclear what the dimension is.
As you know, in particle physics it is common to use eV for particle energy and SI defines 1 eV= 1.602176634×10−19 J
Post by: Aeris on 01/02/2022 13:56:56
"Aeris was the only one to say "Energy" but most of us knew that they were talking only about the potential."
No, I was also talking about the potential. I just referenced energy in my reply so I could simplify my response and make sure that I was on the right track.
Post by: Eternal Student on 01/02/2022 19:41:13
So let's see if there are any questions left over from Aeris.
Let's say, hypothetically speaking, a macroscopic object like a table or a chair undergoes Quantum Tunneling and essentially teleports from one location to another while I'm still looking at it. What would that look like? would the object just pop right out of and then immediately back into existence in the blink of an eye?
Let's just look at a microscopic objects for a moment (before we speculate and generalise to macroscopic objects).
We're going to consider the typical situation with a square barrier potential between two regions of exactly the same potential (region I and region III).
Here's the diagram again:
(https://oer.physics.manchester.ac.uk/QM/Notes/jsmath/square_barrier.png)
We're going to fire a particle toward the barrier (from region I) which has Energy, E < V0 exactly as before and we've already discussed the wave function and shown animated diagrams of this earlier.
Firstly remember that the wave function is not the particle, there isn't anything with properties like a particle until you go looking for it (make an observation to locate the particle). The wave function just indicates the probability of finding the particle at a particular place.
If you go back through the earier discussions and animated diagram you'll notice that the wave function isn't always 0 inside the barrier. There is some time when the particle could be found there.
Now this is where various sources of information (not just PopSci) are going to mess things up or mis-represent what seems to be shown by the mathematics.
The amount of time over which the particle could be found "in the barrier" is controversial. Some sources are going to tell you that it "instantly appears" on the other side of the barrier or just that "it will never be found in the barrier" etc. I don't wish to use too much bad language but this is clearly utter bolderdash.
Just go back and look at the animated diagram carefully if you want to, instead of doing the mathematics. There are times when the real and imaginary components of the wave function are non-zero at a fixed x position inside the barrier. So the square of the modulus of the wave function is not 0 at that time, this is the probability of finding the particle at that given point x in space and given time t. I'm labouring the point here but we just need to be clear that the probability of finding the particle at that position x in the barrier is not 0 for some of the time. The probability of finding of the particle in the barrier does not drop to 0 until some finite non-zero time has elapsed after the incident wave packet hit the barrier. You will find many articles, some textbooks and countless discussions on Quora and similar websites that discuss qunatum tunneling as if it's instant. Some of them are quite interesting and authoritative and discuss problems like the apparent breech of the speed of light (If the particle could move from one side of a barrier of thickness, d>0, to the other side instantly than that is faster than light speed travel). They're all good in their own way except that the fundamental premise they were based on was bollderdash.
Anyway, the mathematics never implied that tunneling would be instant.
Here's one article (from Scientific American) that describes an attempt to actually measure how long it takes a particle to tunnel through a barrier:
https://www.scientificamerican.com/article/quantum-tunneling-is-not-instantaneous-physicists-show/
The details aren't too important, we just need to note that there may actually be some way of observing or measuring tunneling time in practice and it does not seem to be instantaneous. It can be that the particle spends some time in the barrier.
Anyway, we can now try to generalise this to a macroscopic scale: Basically, there's no reason to assume that the quantum tunneling of your chair from region I to region III would be "instant". It's not going to go "poof" from here and just "poof" into existance over there. It is very likely to spend some time "in the barrier" between the two places. I'm not sure what this would "look like" but it could be a lot less impressive than teleporting. Given a piece of space between the start and end point (i.e. a piece of space in the barrier), then the chair could be found there for a short amount of time during the tunneling. I really don't know how to paraphrase this, applying QM to macroscopic objects is always just going to be speculation anyway. It might look like the chair was just moving from the start position to the end position and occupies at least some but possibly all of the positions between those two points in passing. A whole lot less impressive than teleportation, I'm sorry to dissapoint you.
Another long, post, sorry. In summary the only important thing is that Quantum Tunneling isn't instant and may not look like teleportation.
Best Wishes.
LATE EDITING: I probably should make it clear that the time it takes for a particle to tunnel through a barrier is controversial. This was stated originally but it's worth stating again. You should probably make your own investigations. Personally, I'm sticking to what the Mathematics shows.
Post by: wolfekeeper on 02/02/2022 01:07:01
I think also it can happen in even more common and relatively mundane circumstances. If you have a microwave oven where the plastic front of the door is broken, but the metal grill behind it is still intact, it's safe, but if you put your finger near it, the microwaves will jump out of the oven and deliver power to your finger- you will get an RF burn. Don't try this at home kiddies! I believe the microwaves (i.e. the photons) are jumping the gap due to tunneling, but it doesn't happen significantly unless the gap to your finger is relatively small compared to the wavelength and the size of the holes. You can predict it classically with Maxwell's equations, but Maxwell's equations are just QED writ large.
It wouldn't surprise me in the slightest if certain chemical reactions are tunneling also.
Post by: hamdani yusuf on 02/02/2022 09:45:25
In the animation description, it's literally stated that the compared parameter is the energy of the barrier and the mean of wave packet energy. I'm curious what should we interpret these things :Quotehttps://en.wikipedia.org/wiki/Quantum_tunnelling#The_tunneling_problemIf the barrier energy is increased to 40, will there be a portion of the wave packet passes through the barrier?
(https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/E14-V20-B1.gif/330px-E14-V20-B1.gif)
A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.
1. What kind of experiment is best represented by this animation.
2. The amplitude of the wave packet.
3. The distinction between the real part and imaginary part of the wave function (psi).
4. The 90 degrees phase difference between the real part and imaginary part of the wave function (psi).
5. The difference between positive phase and negative phase of the wave function (psi).
6. The number of waves in the wave packet.
Post by: Eternal Student on 02/02/2022 21:02:36
Everything you said seems reasonable. I didn't know about the microwave ovens.
Quantum tunneling isn't confined to the sun, it happens in radioactive decay, I'm not sure off-hand whether all radioactive decay, but certainly where alpha particles are released.I think Halc and evan_au mentioned that earlier. Evan_au mentioned fusion H + H --> He (possibly via heavy hydrogen first) which I think is going to be the more important aspect of what is going on in the sun. Alpha decay is a great example of quantum tunneling but it's only relevant for massive nuceli not the little fellows like Hydrogen in the sun.
Best Wishes.
Post by: hamdani yusuf on 02/02/2022 22:46:48
I'm curious what should we interpret these things :The best I can find is evanescent wave coupling.
1. What kind of experiment is best represented by this animation.
https://en.m.wikipedia.org/wiki/Evanescent_field
I've made my own experiment demonstrating this using microwave frequency a few years ago.
Post by: Eternal Student on 02/02/2022 23:18:43
Well these are all good questions from @hamdani yusuf .
I'm going to try and rush through some answers.
1. What kind of experiment is best represented by this animation.I would imagine the animation was obtained purely on calculation from the Schrodinger wave equation and isn't based on any experiment.
There are numerous experiments that seem to demonstrate quantum tunneling in operation. The first thing that comes to mind are all the Scanning Tunneling Microsocopes (STM) that are in use. If you were just after hard evidence that it works in practice then I would start by Googling that. Someone else has also mentioned electronic components like transistors. I've previously linked to an article in Scientific American that describe some experiments including who did what, where and when so that you could find the original research papers if you were inclined. I'm not really much of an experimentalist myself (you might have noticed). STM 's - they work, they're good and they are the state-of-the-art application of Quantum tunneling.
2. The amplitude of the wave packet.Basically higher amplitude represents an increased probability of finding the particle there. The probability of finding a particle at a given place and time is proportional to the SQUARE of the magnitude of the wave function, ψ. It's conventional to just multiply the amplitude by a constant so as to "normalise" the wave function. So there's nothing really special about the amplitude, it doesn't represent how much Energy the particle had or anything like that. It's been deliberately scaled up or down to "normalise" the wave function.
For a normalised wave function we have the relationship that |ψ(x,t)|2 is precisely equal to (not just proportional to) the probability of finding the particle at position x and time t.
3. The distinction between the real part and imaginary part of the wave function (psi).Surface level answer: Not much.
Medium level answer: There are books and YT videos that discuss how and why complex numbers are important in QM. There are some arguments that QM only works provided the wave function is complex valued and that this also provides some evidence for the existance of complex numbers in nature. Personally, I think that's a ridiculuous set of arguments.
High level answer: You're going to need another thread.
4. The 90 degrees phase difference between the real part and imaginary part of the wave function (psi).Provided the potential V is just a function of position and doesn't vary with time then you get solutions to Schrodingers wave equation that are separable or break into two easy parts: A time dependant component T(t) and a space dependant component φ(x). The over-all wave function is the product of the two ψ(x,t) = T(t) . φ(x). The thing that makes the wave function look like it waves up and down as time passes is of course the time dependant component T(t).
That time dependant component T(t) will be of the form e iωt because Schrodinger's wave equation produces an ODE (in the time derivative) with constant coefficients. Applying Euler's formula you get eiωt = Cos ωt + i . Sin ωt. Now Sine and Cosine are 90 degrees out of phase so most of the complex solutions you generate will have real and imaginary parts that have time dependance like Cosine and Sine repectively.
If you want to look harder and see if there's some deeper significance then we can. I'll try and sketch some ideas here:
As previously mentioned you can scale the wave function with an arbitrary complex number (indeed we MUST do this to normalise the wave function). So we can multiply ψ by the complex number Cos β + i. Sin β for arbitrary angle β since all this will do is rotate it through the angle β in the complex plane without changing its magnitude. So there's no significance in the actual phase of either the real or the imaginary part on their own, you can shift that phase as you please. If there's anything interesting about the phase then it could only be the difference in phase that is of any interest or represents any important property.
There's not a great need to attach some importance to this phase difference because the magnitude of the complex number, ψ, already has an established significance. We already pay a lot of attention to what the overall magnitude of ψ is doing and how that varies with time. If one of the components, say the real part, was showing a rapid oscillation with time then there really isn't a lot of choice about how the imaginary part must oscillate unless we do also see an equivalent rate of oscillation in the overall magnitude of ψ.
I don't know how familiar you are with complex numbers and I'm going to guess that no one wants to see a lot of mathematics here. So unless you're really keen to discuss it further I'll just skip to the conclusion: The phase difference is locked at 90 degrees because that's the only way that you could get rapid oscilliations in either of the real or imaginary part without observing some equally rapid oscillation in the overall magnitude of ψ.
5. The difference between positive phase and negative phase of the wave function (psi).... see comments above... you can rotate the wave function so that you have total freedom to decide if the real or imaginary part is considered to be leading the other one.
6. The number of waves in the wave packet.See details in the Wikipedia page or similar information about "generalised wave packets". See Page 38 of Quantum Mechanics by Franz Schwabl, if by any chance you had the same book as the one I was reading.
In summary, I don't think it's all that important and this post is already too long. All that was important is that you could see some waves. If they (Wikipedia) had tried to construct an even more localised (shorter in space) wave packet or else given the oscillations a longer wavelength by adjusting the Energy, E, so that there were less wave peaks and troughs visible then you wouldn't have seen things as easily in the animations. The number of wave peaks is determined by those things (localisation of the wave packet and Energy E).
Best Wishes.
Oops. I can see I've overlapped with a reply from Hamdani. I don't have any more time to spare, sorry, I'll just hope some of this was useful for now.
Post by: hamdani yusuf on 03/02/2022 02:38:07
I would imagine the animation was obtained purely on calculation from the Schrodinger wave equation and isn't based on any experiment.The follow up questions are then:
- Is the animation a correct representation of Schrodinger wave equation?
- What does the animation try to achieve? Does it simplify the process, e.g. by removing distracting parameters, just like ignoring air resistance to calculate the trajectory of a cannon ball?
- Does it act as analogy by replacing an abstract concept with a more familiar concept, like analogy of electrical current using water flow?
- Does it help us to make a correct prediction of an experimental result?
If none of the above is true, I'm afraid that the animation will only add obstacles for those who try to understand the subject by creating a false conviction of knowledge.
AFAIK, even Schrodinger disagreed with Max Born on how to interpret the value of ψ.
Post by: Eternal Student on 03/02/2022 04:18:46
Is the animation a correct representation of Schrodinger wave equation?No. Only of one one consequence of the wave equation (tunneling through a square walled barrier).
What does the animation try to achieve? Does it simplify the process, e.g. by removing distracting parameters, just like ignoring air resistance to calculate the trajectory of a cannon ball?It shows how a simple wave packet evolves when it is incident on a barrier. It's animated so that temporal changes are shown in real time not in some static graph of something vs. time.
Does it simplify? Not much, I think - but I didn't make it so I wouldn't be sure. Wikipedia made it or someone made it and Wiki acquired it. However, it's already been mentioned that most real life potentials aren't square walled, they may be steep hills but you would usually expect them to be smooth functions not step functions.
Does it act as analogy by replacing an abstract concept with a more familiar concept, like analogy of electrical current using water flow?No, it's still fairly abstract. It's not an attempt to explain something in some different way. It's more like just drawing a graph to show a function like y= x2 (because you could use words to describe the behaviour of y=x2 but drawing a graph is a better way to show it). This graph is animated so the values can be shown changing in real time. The Real and Imaginary part of the wave function shown are the values you get from the solution of the Schrodinger wave equation with that potential. (Don't get me wrong, I didn't make the animation. Overall it looks right and matches diagrams in books but this one is animated so it moves. I haven't checked any values to be honest. They might have deliberately fudged a few values up or down a bit to make the animation a bit smoother or some other thing).
Does it help us to make a correct prediction of an experimental result?Well, it helped me and I would have thought it's useful for other people. If it helps a person understand then they are much more likely to use the theory correctly. You might be confusing "the animation" with the general theory, the animation isn't the theory it's just one graphical representation of one set of results for one situation. It's representative of what happens in similar situations but that is all.
You can't change the parameters like the potential barrier height in the animation - it's a demonstration not an online calculator. Colin2B gave links to an online calculator earlier, there you can enter your own parameters and the appropriate results will be calculated.
If you meant is the general effect known as quantum tunneling useful and does it make predictions: Yes to the first - it's what inspired the creation of Scanning Tunneling microscopes and it does seem to explain how and why they work. Yes to the second, you can make predictions and they can be tested. For example there are some structures like long chains of Gold atoms that have been studied under an STM. You can model the behaviour of the valence electrons in the gold chain quite well with just a 1-dimensional solution to the Schrodinger equation where the gold chain is effectively "the box" ( an infinite height square walled potential well). You get solutions for the electrons in various different states. In the n=2 state you should have a node (a region of constant 0 valued wave function) right in the centre of the gold chain. Using the general theory of tunneling you should then detect the smallest current flow of electrons from the metal chain to the tip of the scanning micro socope when the tip is precisely over the centre of the chain. When you do the experiment, that is what you find.
AFAIK, even Schrodinger disagreed with Max Born on how to interpret the value of ψ.This is true. Quantum mechanics is bizarre and it's frequently said that no one understands it. The Quantum tunnelling effect is just something you get by doing the calculations and seeing what the consequences will be. It's not necessary to understand what ψ represents on some deeper level, or why the wave-function collapse happens, or any of the mysteries and controversies that are frequently attributed to QM. Quantum tunneling just requires you to accept the axioms of QM and calculate.
Best Wishes.
Post by: Eternal Student on 03/02/2022 14:40:07
The best I can find is evanescent wave coupling.I can see your interested in performing experiments. I've now spent some time trying to find an experiment that might be achievable at home with limited equipment.
https://en.m.wikipedia.org/wiki/Evanescent_field
I've made my own experiment demonstrating this using microwave frequency a few years ago.
One of the cheapest and most readily available sources of quantum particles is light, you can get it out of a window although a torchlight or better still a laser pointer device would be better for this experiment. You might be able to do something similar with your microwave genrator but you'll have more trouble "seeing" it, with light you should just be able to see it happening.
The general idea is to fire a beam of light through a medium so that you observe total internal reflection at the boundary of that medium. The light wave should have gone evanescent just outside that medium where it was incident on the boundary. Your going to try and get that to re-enter another block of the same medium before the wave amplitude has dropped off too much, if it works then an ordinary travelling light wave should be re-established in the second block (although with much lower amplitude than in the first block). So, all this is going to involve is having two blocks of glass and trying to put them very close to each other.
I found the details on page 149~150 of An introduction to modern Astrophysics, Carroll and Ostlie, 2nd edition, 2014. I've attached a picture of the relevant section from that book.
Best Wishes.