Video Infoblog: I was wrong about the Heisenberg Uncertainty Principle

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 You've probably heard the Heisenberg Uncertainty  Principle described like this. Measurements knock  

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the state of a particle and disturb it, and  that's why you can't know the position and  

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the momentum at the same time. So for example,  if I wanted to know where this electron was,  

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then I'd have to bombard it  with high energy gamma rays.

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But when I do that Even though it will give me a  pretty accurate representation of its position,  

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it will have knocked this state  so badly that now I have no idea  

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what speed it's going at. But as some of  you may already know, this is completely and  

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utterly wrong. Heisenberg uncertainty isn't about  measurement, it's about the spread of a particle.

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Or at least, that was the version of the  Heisenberg Uncertainty Principle that I  

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was taught when I went to university. But  I can almost guarantee that you haven't  

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heard of the third version of the Heisenberg  Uncertainty Principle, which is called the  

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Measurement Disturbance Relationship, which  has an almost identical looking equation.

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Because, well, this equation was proposed in  1927, the measurement disturbance relationship  

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was only proved in 2003 and it's still  not widely known among physicists. I only  

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learned about it pretty recently. So in  this video I'm going to explain why this  

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version of the Heisenberg Uncertainty  Principle is completely wrong but this  

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version of the Heisenberg Uncertainty  Principle without measurements in it.

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makes a lot of sense and yet how the  measurement disturbance relationship  

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is the Heisenberg uncertainty principle that  you thought you knew but done correctly. It  

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is about how measurements disturb the  quantum state. First let's get into  

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what's wrong about this picture where when we  measure the electron we knock it and that's  

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what causes the uncertainty in the Well,  basically everything about this is wrong.

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In particular, thinking of an electron like  this is the real problem. I do this all the  

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time. It's so fun to draw a little face on  an electron. But electrons are not little  

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balls like this. In fact, they're way more like  waves. Instead of being at one point like this,  

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in quantum mechanics, an electron is in  a superposition of many different places.

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It's kind of spread out. But that doesn't mean  that it's spread out everywhere equally. It  

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might be still centred on one location and have  either more or less spread. So we want some way  

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to kind of quantify where is the electron  and how spread out is it. So that's why  

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we draw this graph. This graph is called the  probability density function for the electron.

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And it gives us an idea of where  the electron is, roughly. So here,  

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you can see that it's mostly kind of around  this point. So it's somewhere on this line,  

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but mainly around here. And we can say  that the spread is kind of this much.  

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That means that the electron is mainly inside  of this region. And so we call this delta x.

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And the way we actually calculate delta  x is we use the standard deviation. But  

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that doesn't really matter. The main idea  is that this is a way to kind of quantify  

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how spread out the electron is. Because if  we had an electron that was more spread out,  

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then this number delta x would  have to become a lot bigger.

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bigger. So delta x is just a way to measure  how spread out is the electron. This delta x  

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is actually the same delta x in the Heisenberg  uncertainty principle and people usually refer  

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to it as the uncertainty in position but I  really hate that terminology because it's not  

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like the electron really is in one of these  places, but we're unsure about where it is.

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In fact, the electron is spread out  so that it's in all of these places,  

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and delta x is quantifying the spread. But now  that we've cleared that up, let's figure out  

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what delta p is. So just like with position, the  electron doesn't have a well defined momentum,  

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so that means that it's going at a superposition  of lots of different speeds at the same time.

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And just like with position, we need some way  of quantifying it. So we're also going to draw  

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a graph for that. This graph is called the  probability density function for the momentum  

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and you can see that it shows the electron is in  a superposition of lots of different speeds like  

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it's going very very slow over here and very  very fast over here and everything in between.

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Mainly its speed is whatever this value is. And,  the majority of its speeds fall into this range,  

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which is called delta p. So this delta p  is just the spread in momentum. Now the  

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Heisenberg Uncertainty Principle says that  there's a relationship between this delta x,  

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the spread in position, and delta  p, which is the spread in momentum.

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It says that if there was an electron that had  an extremely small delta x like this, then this  

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delta x is small. And so, to make sure that delta  x times delta p is still greater than this value,  

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delta p must be bigger. So, the momentum is  greater. is much more spread out. In other words,  

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if we happen to have an electron that  does have a fairly well defined position,  

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then its momentums have to be a  superposition of a big range of speeds.

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The other extreme holds as well. If the  speed is in this relatively small range,  

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then the position has to be in quite  a big range. But notice that none of  

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this is about measurement. It's not  about the electron being buffeted by  

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outside forces and then that's what  changes its state to make it do this.

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Instead it's just about the state of the electron  when it's minding its own business. This is the  

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version of the Heisenberg Uncertainty  Principle that I learned in university  

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and most people kind of encounter it. And so I  was pretty surprised to find out that there is a  

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version of the Heisenberg Uncertainty Principle  that does actually talk about measurements.

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So let's get on to that. Before we go on though,  

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I have an announcement. So for the  last four weeks, I've been running  

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this course as a live course. So every week I set  homework for my students And then we meet and we  

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talk about those homework questions and anything  else they want to discuss about quantum mechanics.

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It has been so much fun. The questions people  ask have been so great and I think, or at least  

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I hope, that the students learnt a lot from it. So  if you're interested in doing something like that,  

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there's more information about the January cohort  of this same class, link is in the description.  

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Okay, thanks.

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Let's go back to the Heisenberg Uncertainty  Principle. The measurement disturbance  

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relationship proposed by Ozama in  2003 looks surprisingly similar to  

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the Heisenberg Uncertainty Principle.  Well actually, that's not true. There's  

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a more general version of the measurement  disturbance relationship which looks like this,  

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but in the where the measurement  is independent of the state.

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So in other words, the way that you're  measuring doesn't depend on already knowing  

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something about the state, which is usually  the case, then it simplifies down to this  

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equation. So for the rest of this video we're  just going to ignore the really complicated  

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version of this measurement disturbance since  most of the time this is the one that applies.

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So yeah, the measurement disturbance relationship  is about how much a measurement will disturb  

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a quantum state. But the reason for that  disturbance isn't just because the particle  

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is getting knocked by external forces. In  fact, this disturbance relationship shows  

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that there is a fundamental limit to how  little you can disturb a quantum state.

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And the reason for that disturbance has  nothing to do with forces or knocking  

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the particle or anything like that. It's to  do with measurement collapse. For example,  

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let's say I have this state which has  this much spread in x and this much  

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spread in p. And let's say that this  is at the limit. So in other words,  

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delta x and delta p are as small as they  can be because they're equal to this limit.

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But what if we now come and measure the  position? So this is my little ruler and  

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I'm going to try and figure out where the  particle is. Well you might expect that if  

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you were to measure the position of this electron  then you should get a range of different values  

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because we said that the electron was in  a superposition of all these different  

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positions but actually what happens is you get a  single result or to put it a bit more precisely.

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If this is the accuracy of our ruler, so it can  only tell if the electron is sort of in here,  

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in here, or in here, then it's not going  to say that the electron is in all three  

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of these places. It's going to say it's  in one of them. Let's say this one. But  

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getting this result fundamentally  changes the state of the electron.

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Now, the electron's state looks like this.  Why did measuring the state change it? Well,  

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we actually don't know. This is called  measurement collapse, and it doesn't happen  

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because we did the measurement really  clumsily and we knocked the electron,  

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and we could have actually done better. It's  something fundamental to quantum mechanics.

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No matter how you do the measurement, once  you know this information about the state,  

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it's forced to change. And so now, this  state has a new delta x and it's much  

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smaller than before. So what has  to happen to the momentum? Well  

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the traditional version of the Heisenberg  Uncertainty Principle gives us the answer.

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Delta p has to get bigger to compensate. In other  words, measuring the position of this electron  

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really did change its momentum, which is exactly  what the measurement disturbance relationship  

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tells you would happen. So this Epsilon x is  actually the error in your measuring device.  

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So here we have this measuring device  that's not really that good, right?

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It can only resolve position up to these levels.  And so the error in this measurement is the sort  

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of size that it can resolve the position, which  is Epsilon x, and in this case is pretty much  

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equal to delta x. So that's the first part of this  equation. It's about the error in x. But what's  

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this? Nu of p is a way to try and quantify how  much the spread in p has changed to compensate.

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So in other words, it's the disturbance in  the momentum. So now the meaning of this  

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equation becomes clear. the less the error  is in x. So for example if I was to use a  

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better ruler that has a finer resolution the  error in x would decrease which would lead to  

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a more peaked version of the position which  would then disturb the momentum even more.

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So that I think gives us the right way to think  about the Heisenberg Uncertainty Principle.  

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Usually people say the better that you know x the  worse you know the momentum as if the particle  

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really had a position and really had a momentum  and you're just trying to discover what that is.  

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But this version of the Heisenberg Uncertainty  Principle says that instead there is a spread  

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of both position and momentum and when you try  and resolve the position by doing a measurement  

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That actually collapses the state and  makes the momentum more spread out.

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So there you go, that's the Heisenberg Uncertainty  Principle. And by the way, if you're interested in  

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signing up to the January version of this course,  the link's in the description. See you soon.