I was wrong about the Heisenberg Uncertainty Principle

You've probably heard the Heisenberg Uncertainty Principle described like this. Measurements knock the state of a particle and disturb it, and that's why you can't know the position and the momentum at the same time. So for example, if I wanted to know where this electron was, then I'd have to bombard it with high energy gamma rays. But when I do that Even though it will give me a pretty accurate representation of its position, it will have knocked this state so badly that now I have no idea what speed it's going at. But as some of you may already know, this is completely and utterly wrong. Heisenberg uncertainty isn't about measurement, it's about the spread of a particle. Or at least, that was the version of the Heisenberg Uncertainty Principle that I was taught when I went to university. But I can almost guarantee that you haven't heard of the third version of the Heisenberg Uncertainty Principle, which is called the Measurement Disturbance Relationship, which has an almost identical looking equation. Because, well, this equation was proposed in 1927, the measurement disturbance relationship was only proved in 2003 and it's still not widely known among physicists. I only learned about it pretty recently. So in this video I'm going to explain why this version of the Heisenberg Uncertainty Principle is completely wrong but without measurements in it. makes a lot of sense and yet how the measurement disturbance relationship is the Heisenberg uncertainty principle that you thought you knew but done correctly. It is about how measurements disturb the quantum state. First let's get into what's wrong about this picture where when we measure the electron we knock it and that's what causes the uncertainty in the Well, basically everything about this is wrong. In particular, thinking of an electron like this is the real problem. I do this all the time. It's so fun to draw a little face on an electron. But electrons are not little balls like this. In fact, they're way more like waves. Instead of being at one point like this, in quantum mechanics, an electron is in a superposition of many different places. It's kind of spread out. But that doesn't mean that it's spread out everywhere equally. It might be still centred on one location and have either more or less spread. So we want some way to kind of quantify where is the electron and how spread out is it. So that's why we draw this graph. This graph is called the probability density function for the electron. And it gives us an idea of where the electron is, roughly. So here, you can see that it's mostly kind of around this point. So it's somewhere on this line, but mainly around here. And we can say that the spread is kind of this much. That means that the electron is mainly inside of this region. And so we call this delta x. And the way we actually calculate delta x is we use the standard deviation. But that doesn't really matter. The main idea is that this is a way to kind of quantify how spread out the electron is. Because if we had an electron that was more spread out, then this number delta x would have to become a lot bigger. So delta x is just a way to measure how spread out is the electron. This delta x is actually the same delta x in the Heisenberg uncertainty principle and people usually refer to it as the uncertainty in position but I really hate that terminology because it's not like the electron really is in one of these places, but we're unsure about where it is. In fact, the electron is spread out so that it's in all of these places, and delta x is quantifying the spread. But now that we've cleared that up, let's figure out what delta p is. So just like with position, the electron doesn't have a well defined momentum, so that means that it's going at a superposition of lots of different speeds at the same time. And just like with position, we need some way of quantifying it. So we're also going to draw a graph for that. This graph is called the probability density function for the momentum and you can see that it shows the electron is in a superposition of lots of different speeds like it's going very slow over here and very fast over here and everything in between. Mainly its speed is whatever this value is. And, the majority of its speeds fall into this range, which is called delta p. So this delta p is just the spread in momentum. Now the Heisenberg Uncertainty Principle says that there's a relationship between this delta x

the spread in position, and delta p, which is the spread in momentum. It says that if there was an electron that had an extremely small delta x like this, then this delta x is small. And so, to make sure that delta x times delta p is still greater than this value, delta p must be bigger. So, the momentum is greater. is much more spread out. In other words, if we happen to have an electron that does have a fairly well defined position, then its momentums have to be a superposition of a big range of speeds. The other extreme holds as well. If the speed is in this relatively small range, then the position has to be in quite a big range. But notice that none of this is about measurement. It's not about the electron being buffeted by outside forces and then that's what changes its state to make it do this. Instead it's just about the state of the electron when it's minding its own business. This is the version of the Heisenberg Uncertainty Principle that I learned in university and most people kind of encounter it. And so I was pretty surprised to find out that there is a version of the Heisenberg Uncertainty Principle that does actually talk about measurements. So let's get on to that. Before we go on though, I have an announcement. So for the last four weeks, I've been running this course as a live course. So every week I set homework for my students And then we meet and we talk about those homework questions and anything else they want to discuss about quantum mechanics. It has been so much fun. The questions people ask have been so great and I think, or at least I hope, that the students learnt a lot from it. So if you're interested in doing something like that, there's more information about the January cohort of this same class, link is in the description. Okay, thanks. Let's go back to the Heisenberg Uncertainty Principle. The measurement disturbance relationship proposed by Ozama in 2003 looks surprisingly similar to the Heisenberg Uncertainty Principle. Well actually, that's not true. There's a more general version of the measurement disturbance relationship which looks like this, but in the where the measurement is independent of the state. So in other words, the way that you're measuring doesn't depend on already knowing something about the state, which is usually the case, then it simplifies down to this equation. So for the rest of this video we're just going to ignore the really complicated version of this measurement disturbance since most of the time this is the one that applies. So yeah, the measurement disturbance relationship is about how much a measurement will disturb a quantum state. But the reason for that disturbance isn't just because the particle is getting knocked by external forces. In fact, this disturbance relationship shows that there is a fundamental limit to how little you can disturb a quantum state. And the reason for that disturbance has nothing to do with forces or knocking the particle or anything like that. It's to do with measurement collapse. For example, let's say I have this state which has this much spread in x and p. And let's say that this is at the limit. So in other words, delta x and delta p are as small as they can be because they're equal to this limit. But what if we now come and measure the position? So this is my little ruler and I'm going to try and figure out where the particle is. Well you might expect that if you were to measure the position of this electron then you should get a range of different values because we said that the electron was in a superposition of all these different positions but actually what happens is you get a single result or to put it a bit more precisely. If this is the accuracy of our ruler, so it can only tell if the electron is sort of in here, or in here, then it's not going to say that the electron is in all three of these places. It's going to say it's in one of them. Let's say this one. But getting this result fundamentally changes the state of the electron. Now, the electron's state looks like this. Why did measuring the state change it? Well, we actually don't know. This is called measurement collapse, and it doesn't happen because we did the measurement really clumsily and we knocked the electron, and we could have actually done better. It's something fundamental to quantum mechanics. No matter how you do the measurement, once you know this information about the state, it's forced to change. And so now, this state has a new delta x and it's much smaller than before. So what has to happen to the momentum? Well the traditional version of the Heisenberg Uncertainty Principle gives us the answer.

Delta p has to get bigger to compensate. In other words, measuring the position of this electron really did change its momentum, which is exactly what the measurement disturbance relationship tells you would happen. So this Epsilon x is actually the error in your measuring device. So here we have this measuring device that's not really that good, right? It can only resolve position up to these levels. And so the error in this measurement is the sort of size that it can resolve the position, which is Epsilon x, and in this case is pretty much equal to delta x. So that's the first part of this equation. It's about the error in x. But what's this? Nu of p is a way to try and quantify how much the spread in p has changed to compensate. So in other words, it's the disturbance in the momentum. So now the meaning of this equation becomes clear. the less the error is in x. So for example if I was to use a better ruler that has a finer resolution the error in x would decrease which would lead to a more peaked version of the position which would then disturb the momentum even more. So that I think gives us the right way to think about the Heisenberg Uncertainty Principle. Usually people say the better that you know x the worse you know the momentum as if the particle really had a position and really had a momentum and you're just trying to discover what that is. But this version of the Heisenberg Uncertainty Principle says that instead there is a spread of both position and momentum and when you try and resolve the position by doing a measurement That actually collapses the state and makes the momentum more spread out. So there you go, that's the Heisenberg Uncertainty Principle. And by the way, if you're interested in signing up to the January version of this course, the link's in the description. See you soon.