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Current time:0:00Total duration:10:11

Video transcript

so when people first showed that matter particles like electrons can have wavelengths and when dubrow's showed that the wavelength is Planck's constant over the momentum people were like cool it's pretty sweet but you know someone was like wait a minute if this particle has wave-like properties and it has a wavelength what exactly is waving what is this wave we're even talking about conceptually it's a little strange I mean a water wave we know what that is it's a bunch of water that's oscillating up and down a wave on a string we know what that is the string itself is moving up and down and it extends through space but it's hard to imagine how is this electron having a wavelength and what is the actual wave itself so physicists were grappling with this issue trying to conceptually understand how to describe the wave of the electron they wanted to do two things they wanted a mathematical description for the shape of that wave and that's called the wave function so this wave function gives you a mathematical description for what the shape of the wave is so different electron systems are gonna have different wave functions and this is this is psy it's the symbol for the wave function so this is psy the size symbol it's a function of X so at different points in X it may have a large value and may have a small value this function would give you the mathematical shape of this wave so is one of the things they were trying to determine but they also wanted to interpret it like what is this wave function even mean so we've got two problems we want a mathematical description of the wave and we want to interpret what is this wave even mean now the person that gave us the mathematical description of this wave function was Irwin Schrodinger so Schrodinger is this guy right here Schrodinger's right here he wrote down Schrodinger's equation and his name now is basically synonymous with quantum mechanics because this is arguably the most important equation in all of quantum mechanics there's a bunch of partial derivatives in here and Planck's constants but the important thing is that it's got the wave function in here now if you've never seen partial derivatives or calculus it's okay all you need to know for our purposes today in this video is that this equation is a way to crank out the mathematical wave function what is this function that gives us the shape of the wave as a function of X and you can imagine plotting this on some graph so once you solve for this sigh as a function of X you could plot what this looks like maybe it looks something like this and who knows it could do all kinds of stuff maybe it looks like that but Schrodinger's equation is the way you can get this wave function so Schrodinger gave us a way to get the mathematical wave function but we also wanted to interpret it what does this even mean to say that this wave function represents the electrons is still strange what does that mean Schrodinger tried to interpret it this way he said okay maybe maybe this electron really is like smudged out in space and it's charge is kind of distributed in different places Schrodinger wanted to interpret this wave function as charge density and I mean it's kind of a reasonable thing to do the way you get a water wave is by having water spread out through space so maybe the way you get an electron wave is to have the charge of the electron spread out through space but this description didn't work so well which is kind of strange Schrodinger invented this equation he came up with this equation but he couldn't even interpret what he was describing correctly it took someone else it took a guy named Max Born to give us the interpretation we go with now for this wave function Max Born said nah don't interpret it as a charge density what you should do is interpret this sy as giving you a way to get the probability of finding the electron at a given point in space so Max Born said this if you find your sy like he said go ahead and use Schrodinger's equation use it get sigh once you have sy what you do is you square this function so take the absolute value square it and what that's going to give you is the probability of finding the electron at a given point now technically it's the probability density but for our purposes you can pretty much just think about this as the probability of finding the electron at a given point so if this was our wave function in other words Max Born would tell us that points where it's zero these points right here with a value of zero there is a zero percent chance you're gonna find the electron there points where there's a large value of sy be it positive or negative there's gonna be a large probability of finding the electron at that point and we could say the odds of finding the electron at a given point here are going to be largest for this value of x right here because that's the point for which the wave function has the greatest magnitude but you won't necessarily find the electron there if you repeat this experiment over and over you may find the electron here once you may find it over here you may find it there next time you have to keep taking measurements and if you keep taking measurements you'll get this distribution where you find a lot of them here a lot of them there a lot of them here and a lot of them here always where there's these Peaks you get more of them then you would have at other points where the values were smaller you build up a distribution that's represented by this wave function so the wave function does not tell you where the electron is gonna be it just gives you the probability and technically the square of it gives you the probability of finding the electron somewhere so even it points down here with the wave function has a negative value I mean you can't have a negative probability you square that value that gives you the probability of finding the electron in that region so in other words let's get rid of all this let's say we sold some Schrodinger equation or we were just handed a wave function and we were told it looks like this and we're asked where are you most likely to find the electron well the value of the wave function is greatest at this point here so you'd be most likely to find the electron in this region right here you'd have no shot of finding it right there you'd have pretty good odds of finding it right here or right here but you'd have the greatest chance of finding it in this region right here so you'd have to repeat this measurement many times in quantum mechanics one measurement doesn't verify that you've got the right wave function because if I do one experiment and measure one electron Boop I might find the electron right there doesn't really tell me anything I have to repeat this over and over to make sure the relative frequency of where I'm finding electrons matches the wave function I'm using to model that electron system so that's what the wave function is that's what it can do for you although if I were you I'd still be unsatisfied I'd be like wait a minute okay that's fine and good wave function can give us the probability or the probability density of finding the electron in a given region but we haven't answered the question what is waving here and what exactly is this wave function is this a physical object sort of like a water wave or even an electromagnetic wave or is this just some mathematical trickery that we're using that has no physical interpretation other than giving us information about where the electrons gonna be and I've got good news and bad news the bad news is that people still don't agree on how to interpret this wave function yes they know that the square of it gives you the probability of finding the electron in some region but people differ on how they're supposed to interpret it past that point for instance is this wave function the wave function of a single electron or is this wave function really the wave function of a system an ensemble of electrons all similarly prepared that you're gonna do the experiment on in other words does it describe one electron or only describe a system of electrons does it not describe the electron at all but only our measurement of the electron and what happens to this wave function when you actually measure the electron when you measure the electron you find it somewhere and at that moment there's no chance of finding it over here at all so does the act of measuring the electron cause some catastrophic collapse in this wave function that's not described by Schrodinger's equation these and many more questions are still debated and not completely understood that's the bad news the good news is that we don't really need to understand that to make progress everyone knows how to use the wave function to get the probabilities of measurements you can have your favorite interpretation but luckily pretty much regardless of how you interpret this wave function as long as you're using it correctly to get the probabilities of measurements you can continue making progress testing different models and correlating data to the measurements that people make in the lab now I'm not saying that interpretations of this wave function are not important people have tried cracking this nut for over a hundred years and it's resisted maybe that's cuz it's a waste of time or maybe it's cuz the difficulty of figuring this out is so great that whoever does it will go down in history as one of the great physicists of all time it's hard to tell right now but what's undebatable is that for about a hundred years now we've been able to make progress with quantum mechanics even though we differ on how exactly to interpret what this wavefunction really represents so recapping the wavefunction gives you the probability of finding a particle in that region of space specifically the square of the wave function gives you the probability density of finding a particle at that point in space this almost everyone has agreed upon whether the wave function has deeper implications besides this people differ but that hasn't yet stopped us from applying quantum mechanics correctly in a variety of different scenarios