 Hi, when you say quantum mechanics is someone, one of the things they first think of is usually the Heisenberg Uncertainty principle, but it's also one of the concepts that people misunderstand the most. In quantum mechanics, remember, everything's a wave, and so it doesn't have one perfect position unless all the wave is piled up into one place. And also because different parts of the wave can move around, it doesn't necessarily have one perfectly well-defined velocity or momentum either. It turns out that if you multiply the spread in momentum, the variance in momentum, times the spread in position, i.e. the variance in position, if you multiply those two variances together, then those can't get smaller than h bar on two. So what that means is you can have a wave that has a really well-defined momentum, but if you do, then it must be really spread out. Conversely, if you have something that has a really well-defined position, then it must have a very large spread of momentum, and so if you look at what happens next, it's going to blow up. And the way people talk about this, and indeed the way Heisenberg originally talked about this, is in the context of trying to measure something. So Heisenberg talked about trying to measure something microscopic. If you try to measure something, you have to interact with it in some way. If you don't ever interact with something, you never get any kind of idea of how it's behaving. And so he talked about cases where if you're trying to measure the position of something, then you might interact, throw light at it in some way, and the light bouncing off of it will change its momentum, it will give it a kick. And that's true, but the problem with that approach is that it tends to make you continue to try and think of clever ways you might try and measure something without giving it kicks. Whereas in fact, the Heisenberg concerning principle is not about how hard it is to measure something, but about the properties that the thing can have in and of itself. So remember that everything in quantum mechanics is described by a wave, whether it be light or matte or anything else, and there's a relationship between the momentum of that wave and its wavelengths. In other words, the actual wave that has a perfectly well-defined momentum has a perfectly well-defined wavelength. In other words, it's a perfect sinusoid, cosine or sine, with some phase going along in space. And so we have a perfectly well-defined wavelength here. So something that has a perfectly well-defined momentum has an uncertainty momentum that's essentially zero. But you can also see that this wave doesn't ever dip down, so it's just as likely to be a million miles that way as it is to be a million miles that way. So the uncertainty in position is effectively infinite. And so we can make one of these quantities very small, but only the cost of making the other one very, very large. Now contrast that to the wave that has a very well-defined position. So it's zero for most of space, and then it has some finite value and then is mostly zero everywhere else. And so you can see that this wave function has a fairly well-defined position, and so its uncertainty in position might be something like that big. So this number here is very small. Unfortunately, it doesn't now have a well-defined wavelength. In order to make this wave here, you have to add a lot of different kinds of sine waves together, so you get a lot of different kind of momentum. And so it has a large spread of momentum. So we've got a small spread in position, and we get a large spread of momentum. And so there's this trade-off between the spread in position and the spread in momentum. And so another way of writing Heisenberg's uncertainty principles is to say a wave that has a well-defined momentum just doesn't look like a wave that has a well-defined position. So this equation here is a graphical version of Heisenberg's uncertainty principle. Now it's not just momentum and position that are related in this way. In quantum mechanics, there are lots of quantities that have an uncertainty principle relationship between them. And one of the more commonly quoted ones is actually energy and time. And in terms of the underlying mathematics of quantum mechanics, this is a slightly unusual one because why you can talk about how spread out a wave is, it's pretty hard to talk about how spread out the time a wave is. In the absence of the object being described by this wave being created or destroyed, then the time that it exists is essentially all time forever. And so what exactly is this equation trying to tell us? Well in quantum mechanics, the energy of a wave determines how it changes. So in Schrodinger's equation, the rate of change of the wave function is proportional to the energy of that wave function. And so if you have a wave with one perfectly well-defined energy, it might still have a spread in momentum, it might still have a spread in position, but it doesn't change. An example of that might be something sitting at absolute zero, it might be an atom sitting in its ground state. And so you have a single wave that's static for all time. In order to have something change, so for changes to happen, so a measurement, energy transfer or something like that, the energy difference, the spread of energies in that wave defines how fast that process goes. And so there's a relationship between the spread in energies and the time scale of any process.