Where does the quantum world begin? Why do we have a hard time fundamentally understanding quantum systems? What connects quantum and classical physics? I discuss these questions and more in today’s Ask a Spaceman!
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EPISODE TRANSCRIPTION (AUTO-GENERATED)
Let's say you wanna break out of prison, and I'm not gonna get into how you ended up in prison in the first place. That is the subject of an entirely different podcast, certainly not ask a space man. Right now, we are focusing on getting you out. What are your options? You don't wanna be in prison.
One, you could go on parole, be released for good behavior. Fair. Two, you could dig a hole. Good luck. Three, you could steal a key from the guard because I'm sure that's still a thing in the twenty first century.
Maybe I could bake you a cake, and I could put a saw inside of a cake. And I've only ever seen this in movies, and even in the movies, it doesn't work. Does this actually work in real life? I don't know. Okay.
It seems your options are limited. What about quantum mechanics? See, we've all heard about quantum mechanics. We've heard about this funny thing called quantum tunneling. Maybe you could just, you know, quantum mechanically tunnel outside of the prison.
Why does this sound ridiculous? Why does it sound ridiculous for you to wait inside of a prison and then one day spontaneously be outside the prison? You know, just like an electron does, quantum tunneling wherever it pleases. This sounds ridiculous because you and me, as far as I'm aware, are not quantum things. We're classical things.
We're we're bound by the rules of classical physics. And in classical physics, quantum tunneling doesn't exist. It's impossible. It's a joke. If you don't have enough energy to cross the barrier, you simply never cross the barrier, period.
End of sentence. End of story. Why are we even still talking about this? Why is this even a thing? In classical physics, like, if there's a wall and I'm not strong enough to break down the wall, I will never break down the wall.
If there's a hill and I don't have enough kinetic energy to get to the top, I will never get to the top and see what's on the other side. Period. Done. How hard is this, folks? Turns out it's pretty hard.
This is true in classical physics, but we live in a quantum universe. At some level, which we'll get into, the classical rules of how the world works, including rules like if you don't have enough energy to cross a barrier, you don't cross the barrier, don't apply. And there's all sorts of weird stuff happening way down there in the quantum world, in the subatomic world, in the tiny world. In the tiny world, the rules that we're used to, the rules that make sense don't apply. There's a new set of rules, and they don't make sense because we grew up in a world where they shouldn't make sense.
If we grew up fully aware of the quantum world, if we've been aware of quantum mechanics for thousands of years, if we evolved as humans in a quantum world, then it would make sense because those would just be the rules. And we'd be astounded by classical physics. Like, what do you mean? If you don't have enough energy to cross the barrier, you don't cross the barrier. What?
Come on. Quit putting a straightjacking on me. But but this leads to a very, very fundamental question. It's the question that's really I'm gonna circulate around for this episode. I've done a few episodes on quantum mechanics.
This one is about the boundary between the quantum and classical world in a very fundamental question. If we fundamentally live in a quantum world, if the rules of how things work down there are all quantum, and I'll explain a little bit how quantum tunneling works. How come you can't break out of prison by simply waiting? You'd have to wait a long time. Like, technically, it might be possible according to the rules of quantum mechanics for you to break out of prison, but you don't.
Nobody counts on this. Even the sneaking a saw into a cake is preferred over this idea. How come the quantum rules don't apply to the macroscopic universe? So to dig in to dig in, let me talk about why an electron has not so much trouble breaking out of prison. You put an electron in a box and you double check.
You're like, yep. That electron is most definitely inside the box. And then you turn around. You do something. You make a sandwich.
You watch a episode of of TV, whatever. You take a nap, and you come back. Electron's gone. It's outside the box. You're like, wait.
Wait. Wait. Wait. Wait. Wait.
I knew the energy that the electron had. It was based on its temperature, maybe some kinetic like, I I know I know how much energy the electron had. I know how thick the walls are. I don't see any holes. It didn't jump out.
How did it get out? It tunneled. It quantum mechanically tunneled outside of the box. And here's the thing. It can tunnel back in.
You're like, okay, Electron. Fine. You can be outside the box. Whatever. No big deal.
You go back to your sandwich, go back to your TV show, go back to your nap, come back, and then the electron's back inside the box. Doesn't care about the wall. Doesn't care about the boundaries. And it doesn't care because an electron is not a classical thing. Electron is not a tiny little ball.
If I put a billiard ball in a box, every time I open the box, I'm gonna find the billiard ball inside of it. Not true of an electron because an electron is not a classical thing. A billiard ball is a classical thing. A person in a prison cell is a classical thing. An electron is a quantum thing, and quantum things have different rules.
One of the rules that quantum things have is that they're not always particles, or you can't always use a particle description to describe it. It's not a ball. It's not a thing. It's something else. It's a particle and a wave simultaneously.
The wave, this wave like property that we ascribe to subatomic particles, is a wave describing where you might find the electron the next time you look. To imagine this, give you a little visual analogy, and I'm I've talked about this in previous episodes. Here we go again because it's it's it's this is it. This is it. Like, this is quantum mechanics.
Imagine waves on the surface of water, like the ocean waves or ripples in a pond, and you see and you freeze frame it like. You see all those ripples on the pond. If those were waves of probability for an electron, then the waves would describe where on that pond, where on that lake, where on that surface I might find the electron the next time I look. So places where the waves are high, the big ripples, likely places to see an electron. In places where the waves are low, the dips, the troughs, don't even bother looking.
Very small chance of spotting an electron there. That's what these waves of probability mean. We call them waves because they act like waves. Wave equations describe them. It's all wavy.
Where we might find an electron is described as waves. And there are places where we're likely to find the electron and places where we're not so likely to find the electron the next time we go looking for it. Now if you put up a wall, put it in a box, The wall doesn't cut off the wave. It's not necessarily a barrier. It doesn't just, like, stomp out the wave.
The wave still exists. The wave is still there, but it does modify the wave. It does change the wave. It reduces the strength of the wave outside of the box. If you have a really, really thin barrier, well, then the wave isn't gonna be modified very much.
If you have a really thick or tall barrier, well, then the wave is gonna be diminished a lot. But the wave still exists because technically here we go. You're right for this one, guys. Technically, the wave extends to infinity. If I have an electron and it's right there, next time I look, it could be anywhere in the universe, literally anywhere.
Now the chances of it appearing in the Andromeda galaxy are somewhat slim, where I'm not really gonna worry about it. But still, there is a chance that the next time I look for the electron, it will show up in the Andromeda galaxy because its wave extends throughout the universe just with ever decreasing probability. The presence of a wall or a barrier modifies the wave, diminishes the wave, but it can't get rid of the wave. Nothing gets rid of the wave. And so if the wave describes where the electron might be the next time I look for it, all the barrier does is modify the wave, change it a little bit, then there's still a chance outside the box that I'll see an electron.
In fact, from the electron's perspective, if it had a perspective, let's go there, there's no such thing as inside or outside the box. There's just the wave. There's just the wave of where I might decide to be the next time I need to interact with something or someone needs to observe me. Next time I need to do something, I'm gonna be somewhere along this wave. And, oh, look.
There's a barrier. Interesting. All that does is modify my wave. So quantum tunneling is a part of the quantum world because at the quantum level, everything's described by waves and particles simultaneously. The wave describes where I might find the electron, and I go look for the electron.
And sure enough, there's an electron that looks like a little particle. If I insist on particles, on electrons being particles, on electrons being tiny little hard balls, then the quantum tunneling doesn't make sense ever. I need to include the wave description in order to make this work. I need to go all quantum. I need to go all in on the quantum world in order for this to make sense.
This is hard to think about. Trust me. This whole wave particle duality thing where you need both a wave description and a particle description together simultaneously in order to describe the quantum world, this is hard. One of the first people to point out just how hard this is is wonderful physicist by the name of Niels Bohr. Niels Bohr, one of the founders of quantum mechanics, operating in the early and mid twentieth century, having lots of deep, deep, deep thoughts, and he had some very deep thoughts about quantum mechanics.
And his deep thoughts were almost precursors, almost bedrocks to quantum theory, to this wave particle idea in the early twentieth century. And I've talked about this on other episodes. You know, we're doing all sorts of experiments. We're discovering spin and the double slit experiment and radioactive decay and the photoelectric effect and and Compton's like like, all this cool stuff that made no sense. And we were coming up we as in, you know, smart physicists, not not me personally, we were coming up with just, like, sketches, like hacks here and there, like Planck's constant.
Was it like a hack to explain some experiment? And slowly, over the first couple decades of the twentieth century, physicists were starting to get a handle on it and starting to have some sort of overarching concepts, some great like, some unifying ideas behind this mechanics of the quantum, a k a quantum mechanics. And together with that, together with the mathematics, people were having ideas. They were talking about it, like, philosophically. Like like, I how should we think of the quantum world?
And Niels Bohr introduced this concept of complementarity. He pointed out something very interesting. In the early twentieth century, we Einstein had already figured out special relativity, and special relativity hinges on this finiteness of the speed of light. Speed of light is constant, and so there's always this relationship between space and time. You can't separate space from time.
You can only think about unified space time together. They are complementary concepts. They they're they're joined at the hip. There's only one space time. We point out, like, look.
Look. There's that constancy of the speed of light. It's a finite number that unifies space and time. There's also another constant at the other end of the scale, Planck's constant, the quantum constant, the number that describes quantum mechanics as discovered by Max Planck. It's a tiny number, but it's there and it's finite, and this constancy joins two other things.
It joins the wave and the particle nature. The constancy of the speed of light joins space and time. The constancy of quantum, of Planck's constant, joins the wave and the particle. It joins other things too. It joins the behavior of a system with its measurement.
You can't measure something without interacting with it and affecting it. It joins, say, position and momentum or energy and time. I'll talk more about those. If if if you'd like me to do someday an episode on the Heisenberg uncertainty principle, then, just ask because I'd love to dig in more into the Heisenberg uncertainty principle. Heisenberg uncertainty principle, which you've probably heard of, is a subset of Niobr's complementarity principle, which you probably haven't heard of.
It sits as a as a tiny corner of it, but Bohr's complementarity principle goes bigger than the Heisenberg uncertainty principle. It's not just position and momentum or energy and time. It's particle and wave. It's observer and observee. They're all joined at the hip, and they're joined at the hip because there is a fundamental constant of nature at the quantum level, just like space and time are joined at the hip because there's a fundamental constant speed of light.
Bohr's point, besides laying a founding bedrock for all of quantum mechanics, is that it's hard to think about the quantum world. It's hard for us to wrap our brains around the quantum world because it operates on a different baseline. It operates under a different set of principles. Our classical world, there are particles and there are particles and there will always be particles, and there are waves and they are waves and there will always be waves, and a particle is not a wave and a wave is not a particle, period. End of story.
You will not get out of prison, period. End of story. Quantum world's different. Quantum world things are paired. Properties are paired together.
One of the pairs is particles and waves. There's no such thing in the quantum world as just particle or just wave. There's only worticles. Paves, still working on it. A better name.
We're just gonna call it particle wave duality or wave particle duality, your choice. Complementary pairs, also known in the jargon as conjugate pairs. The quantum world exists in these pairs, and you can't ever know everything about a system, a property. The more you know about one, the less you know about the other. The more you insist on the wave like nature, the less you know about the particle like nature, and the more you know about the particle like nature, the less you know about the wave like nature.
It's easy for an electron to break out of prison because it lives in a quantum world. It's hard for you to break out of prison because you don't because you live in a classical world. But where's the boundary? Somewhere between electrons and you, it feels like there should be a line. There should be a say, okay.
Okay. Okay. Below this, quantum rules, pairs, conjugate, complementarity, all the good stuff, wave particle duality. Above this, classical physics, electrodynamics, Newtonian physics, special relativity. It's all good.
But where's the line? Where's the line between the quantum and the classical worlds? Niels Bohr, same Niels Bohr, was one of the first to articulate, to talk about a potential boundary between the quantum and the classical world. Because, obviously, this question comes up like, okay, if the quantum world is gonna operate under a different set of rules, how come those rules don't apply up here? Why do we have different rules?
In the early twentieth century, physicists knew that something fishy was going on in the quantum world. They hadn't yet developed that full treatment. They'd have to wait till 1925. And it it's like in this in this time frame, it's like they were starting to figure out the ingredients of how to make a pizza, but they hadn't yet invented the pizza. Like, they had figured out some vague recipes for dough, some they knew that tomatoes were involved in some form.
Dairy was definitely a part of the story, but they just didn't know, like, how to put this together and make a decent pizza. Niels Bohr was one of the first people to try to make a pizza. It wasn't perfect. The dough was really spongy. The sauce was too salty.
The the cheese wasn't cooked. He put it on after it baked. Like, it's technically edible, but but it wasn't really satisfying, and it's not something we would really call a pizza today. This pizza that he had, this imperfect pizza, this imperfect model of the quantum world was a model for the atom. And his model for the atom contained two essential features that would be vital to the future of quantum mechanics, to to the full pizza.
It's like he got he nailed two parts of the process. At the time, people had figured out through various experiments that atoms were made of a small, heavy, dense nucleus that was positively charged surrounded by very tiny, very light particles called electrons. And we had recognized that it was some sort of quantum system, mostly because only certain frequencies of light would be emitted from atoms. If you heat up atoms, you only get certain wavelengths of light. We had known this for a long time, but wasn't until the early twentieth century that we realized that there was something fishy in quantum going on.
The first feature that Bohr introduced into his model, the atom, was that it was quantized. He had the electrons orbiting around the nucleus like planets orbiting the sun, but the electrons were allowed to only be in certain orbits. Like, they couldn't just float around randomly. They couldn't be at any old random distance from the nucleus. They had to be either here or here or here or here or here.
That's quantum mechanics. Right? Like, you can only do this one or this one or this one. You know in betweensies, no havesies, just this, this, this, or this. And and to describe the level of where in the orbit the electron can be, it could be a close one, medium orbit, a far orbit.
He had numbers. He's like, okay. This is essentially, I'm I'm bringing down, like, okay. Orbit number one, orbit number two, orbit number three, orbit number 47. You can't have orbit number point five.
You can't have orbit number 1.34. You you it's just one, two, three, four. These are called quantum numbers. Any numbers that we use to describe a quantum system are conveniently enough called quantum numbers. So his first feature of the model of the atom that would prove useful for later theories of quantum mechanics was actual quantization built into the atom itself.
The second feature was that as the quantum numbers got big, as the electron got further and further away from the nucleus, if you put it in a further and further orbit, it looked more and more like a classical system. If you put an electron far enough away from a nucleus, then all the quantum mechanics don't really matter anymore. It just looks like an electron orbiting a nucleus and doing all its normal electromagnetic stuff. If it got close, if the quantum numbers got low and it got in a low orbit, then it was totally governed by quantum mechanics. But far away, at high energies, at high quantum numbers, it looked more like a classical system.
This is what's called the correspondence principle. The correspondence principle offers a way of translating from the quantum world to the classical world, And it first appeared in Neil Bohr's model early model of an atom. Like, yes, it's quantum, but when the quantum numbers get big, we can kinda sorta ignore the quantum bits and just treat it like a classical system. There's a smooth transition from purely quantum to purely classical, and here's a way to do it. Eventually, Niels Bohr and a bunch of other smarty folks figured out quantum mechanics, and this correspondence principle served as a cornerstone.
It served as a cornerstone because the actual mathematics of quantum mechanics are very broad. It's just it's just very generic mathematics that don't really tell you a lot or can tell you a whole bunch of different things. We need a way to pick out which possible mathematics could actually describe physical systems and which were just fun mathematics. And the correspondence principle helps pick the physical theories. Like, if you if you have a whole buffet of pizzas of all sorts of different arrangements or there's all sorts of different recipes, You want the recipes to actually generate a pizza that you'll like, that will taste good.
Those are the physical ones. Those are the ones we wanna use to make predictions and understand the universe. But there's a whole bunch of ingredients. There's a whole bunch of recipes for pizzas that make pizzas that are technically pizzas but also don't taste good. The correspondence principle helps us pick the tasty pizzas.
Just the right balance, crunchy yet spongy crust, rich sauce, creamy cheese, like like, the whole deal. The correspondence principle helps us pick which pizzas taste good, and the pizzas that taste good are the ones that are good at describing physical reality. So if you wanna make a quantum theory, fine. You go right ahead. Have a nice day.
Do whatever you want. But according to the correspondence principle, whatever rules you develop for the quantum world, whatever recipe you have to make a pizza that describes the quantum universe, those rules must dissolve, for a better word, when you bring it up to the macroscopic world. If the quantum numbers get large, the energies get large, then the quantum mechanical predictions must give you the exact same answer as your normal typical classical nineteenth century, everybody has a suit and a beard and or mustache physics. Like in the atom, electron close to the nucleus, totally quantum mechanics, own set of rules. You put that electron far away from the nucleus, man, you better be able to reproduce electromagnetism as if the quantum world didn't apply.
This transition happens in the quantum world through what are called expectation values. I know I'm throwing a lot of jargon at you. It's it's but welcome to quantum mechanics. It's like all jargon all the time, and none of it makes any sense. You You see, quantum mechanics, another feature of quantum mechanics, like, you know, this whole wave particle duality thing, you never know exactly where the electron's gonna be, but you can describe it probabilistically.
I can tell you where the electron might be. Like, yeah, you got pretty good odds of, the electron being inside the box, but 10% chance of it being outside. You know, I can describe things probabilistically. If you give me a million electrons in a million boxes, I can tell you what, say, 900,000 of them are gonna do, plus or minus. Right?
I can I can give you probabilities? And the probabilities, the average behavior of, say, a million electrons must be the classical result. Here's here's another example. Here's another example. Let's say I throw a ball, whoop, or I throw an electron because it's tiny and it's a quantum mechanical.
It might follow one path, it might follow a million different paths because quantum mechanics, who cares? It's all wave functions and probabilities. Alright? It might follow like one path, it might jump, it might loop around to the Andromeda Galaxy, there it is again. Like, it it might go under use my feet, and it's in the dirt and pop up.
Like, it could go in a million different places. If I throw a million electrons, then the average electron will behave as if I just threw a baseball. The average result of a quantum system must be equal to the classical result. That's how this correspondence happens. That's how this connection actually happens.
You do recover classical physics out of quantum mechanics, but only on average. But this means that there's no hard and fast boundary between the quantum and classical worlds. It's a smooth transition. On one end, it's definitely quantum. On the other end, it's definitely classical.
There are some rules in place to connect the two, like the correspondence principle, like these expectation values, like the average results of many, many different experiments, that's about it. That's about it. Typically, classical physics, if you wanna be rough about it, should apply when the quantum numbers become large. Like, here's another example. I know I'm throwing lots of examples at you because this is a pretty fishy concept.
Trust me. Like, this is a a very squirrely, fishy, jello y concept. This boundary between the quantum and the classical worlds. Here's another example. Angular momentum is quantized.
If I have a certain mass, it can't spin at any old rate it wants to. It can't. It just can't. It fundamentally can. It's a quantum thing.
Angular momentum is quantized. It can spin with one level, like, one speed, and then it can double that and quadruple that or three times it, four times it, five times it, six times it, etcetera, etcetera, etcetera. It can't go one and a half. It can't go three and a half. There's only certain strengths of spin or certain speeds it can have.
But up here in the classical world, the spins are so big. The angular momentum, like me moving my arm around or spinning in a circle, that angular momentum is so huge compared to the fundamental quantum of angular momentum that my quantum number to describe it is, like, eleventy quadrillion. And the difference between eleventy quadrillion and eleventy quadrillion in one is very, very, very tiny. And so to me, everything looks classical and smooth, and I can have any angular momentum I want when really it's a quantum thing. That's one way to look at it.
Another way to look at it is that when the wavelength gets very, very short. That's when the quantum world starts to disappear. Remember, the quantum world, you have to care about both particles and waves, this correspondence, this conjugate, these pairs of properties. You have to care about both the wave and the particle. But if the wave part is really, really, really, really tiny, you can basically ignore it and treat it like a particle.
So like an electron, you can't ignore the wave part because it's too it's so small and its wavelength is so long. You have to include the wave part. But you in a prison cell, you are technically, in some sense, a quantum mechanical object, but your wavelength is so tiny that I can basically ignore it. That, yes, reality has this dual paired nature of both waves and particles, but your wave nature is so suppressed. I can pretend you're a classical system.
That's one way to look at it. This does introduce yet another puzzling question. Just how much should you contribute to Patreon? Go to patreon.com/pmsutter to learn how you can contribute to keep this show and all of my education outreach work going. Maybe it introduces another puzzling question.
Maybe not not necessarily that one. Do we live in a quantum world? Do we live in a quantum world? An electron has this dual nature of particles and waves that we can't ignore. You in a prison cell here's the question.
Here's the question. Is it that you and a prison cell also have the dual nature, except we can ignore one side of that nature just for convenience sake, or do you fundamentally have a different nature? Do the rules of quantum mechanics apply up here in the macroscopic world, but just parts of them go away or dissolve or the quantum numbers are so large or the wavelengths are so short that the quantum just kinda doesn't matter anymore and we can ignore it? Maybe not. Maybe.
Maybe. Maybe not. Up here, classical rules apply. Electromagnetism, Newtonian gravity, air drag, all that stuff. It's a classical world up here.
You might be tempted to say, like I just said, like, okay. No. No. No. When when quantum numbers get big or wavelengths get short, then the quantum nature is just suppressed.
The dual nature is suppressed, and you just have this this classical world again. But the big problem is that the actual mathematics of quantum mechanics don't do that. The actual mathematics of quantum mechanics only produce the classical results on average. So you could rightly argue, let's say me throwing a ball or you sitting in a prison cell is not a quantum mechanics problem. There is no quantum mechanical theory to describe what is happening.
Quantum rules have nothing to do with it. It's just classical rules. I can't apply the quantum rules to break me out of prison because I'm fundamentally a different thing. This was Neil Bohr's argument when he was talking about complementarity. I didn't bring it up at the beginning just for fun.
He argued that as an observer, like, here I am, an observer. I'm a classical thing, very much classical thing. How can I possibly understand the quantum world? The quantum world is hard to understand because we are a classical thing. The quantum world relies on these pairs of complementary properties like waves and particles, position and momentum, and the classical world simply doesn't, and I am a classical thing.
So, of course, the quantum world is not going to make sense because it's almost literally a different world, but maybe not. Maybe our classical world is just an extension of the quantum one. Maybe it's not quantum rules over here, classical rules over here. Somewhere in between is kinda sort of gray zone, but there is this transition between, and there are some rules that we apply to make sure that transition can happen, like the correspondence principle. That's a rule to ensure that at some point, the quantum world and the classical world link up, even though we don't fully understand that linkage.
Bohr is arguing that that link up is just artificial. It just exists in our heads so that we can make the mathematics and have make progress, but the quantum world is fundamentally different than the classical world. There is another equally valid viewpoint that I was talking about with, like, the the quantum numbers and the wavelengths that no. No. No.
We really do live in a quantum world up here. Like, there's a little bit of quantum in you and me. We just don't get to notice it or appreciate it much. And then our classical world is just an extension of the quantum world, not separate from it. What is the answer?
We don't know. Physicists debate it back and forth, have for a hundred years, probably will for another hundred at least. Do we live in a quantum world? We don't know. Do the quantum rules apply up here in the classical macroscopic world, or is it a fundamentally different set of rules?
Does it matter? Thank you so much for listening. Hey. It's been a while since I've promoted an AstroTur, and we've got another one. This is a really, really fun one.
One. The Astro tour cruise that we did in 2018 was a blast. Went to The Caribbean. We saw mine ruins. We did stargazing off the deck.
Hard to describe how much fun we had. So we're doing another cruise. Go to astrotours.co. Astrotours C o, and there'll be a link there, the sea and the stars. It's gonna be another fun cruise in The Caribbean, going to Mayan ruins, stargazing off the deck, eating food, and doing show recordings.
It's gonna be so much fun. Registration window is relatively short because we need to get bookings in early. Deposit is some very small number, like a hundred dollars each to get your name on the reservation list. Go to astrotours.co, and you'll see it right there. The name of the trip is seeing the stars.
It's gonna be September of twenty twenty, and it's gonna be a blast. I promise. Thanks to Roberts l on Facebook, leaving s on email, at GeekSquared on Twitter, James w on email, Benjamin t on email, Newport Flote on Instagram, at Smaddy Wood on Twitter, and at Maria a on email for the questions that led to today's topic. And, of course, my deepest gratitude to all my Patreon supporters, especially my top ones this month, Matthew k, Helga b, Justin z, Justin g, Kevin o, Duncan m, Corey d, Barbara k, Nooter, Drew, Chris c, Robert m, Nate h, Andrew f, Chris l, John, Elizabeth w, Cameron l, and Nalia. Go to patreon.com/pmstarto learn how you can keep this show going.
And don't forget, leave a review on iTunes if you can. That really helps boost visibility. Go to aska spaceman Com. Drop me some questions. You can email askaspaceman@gmail.com.
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