 We've seen that quantum mechanics tells us it's possible to know the magnitude of an electron's angular momentum and its projection along one direction in space, so we can only measure one of the three components of L. This sounds a bit like the uncertainty principle, according to that if we know position precisely then we can't know anything about momentum and vice versa. The uncertainty principle was first presented by Werner Heisenberg and it's sometimes called the Heisenberg Uncertainty Principle. In the early years of quantum mechanics people struggled to understand the full implications of this. One line of thought was that maybe all physical quantities really have precise values, as in our macroscopic world it's just that there's a problem with measuring them. After all in the case of angular momentum we've seen how even in classical physics the very act of making a measurement can change the properties of the system being measured. This is often called the observer effect. Suppose we have a cup of coffee and we want to know it's temperature, so we stick a thermometer in it. Well, if the thermometer is too big it will absorb considerable heat from the coffee and so change the very temperature we're trying to measure. A solution would be to get a thermometer small enough that we can safely ignore the heat it absorbs from the coffee. But in quantum mechanics we don't have this option. You can't exactly build an instrument smaller than an electron. Heisenberg's microscope is a thought experiment that illustrates the relation between the observer effect and the uncertainty principle at the atomic level. Suppose we have a particle moving along the x-axis with some momentum P. We fire a photon towards it. The photon has wavelength lambda, hence momentum H over lambda. Now assume the momentum of the electron and the photon are precisely known. We're going to determine the electron's position by imaging it with a lens. One theory tells us that if at the electron the lens covers an angle theta then the position of the electron can be imaged with the resolution delta x where delta x equals lambda over theta. We can make delta x smaller by making lambda smaller and or by making theta bigger. But when we detect the photon all we will know is that it has passed through the lens. It might have traveled towards the right in which case it will have an x component of momentum equal to H over lambda times theta over 2 or it might have traveled towards the left in which case the Px component is the negative of that. The uncertainty in the photon's x component of momentum is the difference of these values or H over lambda times theta. Since momentum is conserved in the interaction of the electron and photon any uncertainty in the photon's final x momentum implies the same uncertainty in the electron's final x momentum. Thus H over lambda times theta is also the uncertainty in the electron's final x momentum. Multiplying delta x and delta Px we obtain the uncertainty principle. Even if we knew the electron momentum exactly to start the very act of measuring its position introduced a momentum uncertainty. And the more precisely we try to determine position by using a smaller wavelength photon or a larger lens we end up introducing a larger momentum uncertainty. This illustrates how even if the electron has a precise momentum and precise position we wouldn't be able to determine both with arbitrary precision. In fact as we saw in video 3 in the wave particle duality picture the concept of simultaneously well-defined position and momentum doesn't even make sense. Heisenberg described how difficult it was to come to terms with these concepts. He kept asking himself can nature possibly be as absurd as it seemed to us in these atomic experiments? He concluded that it's natural for us to ask what happens really in an atomic event but quantum mechanics tells us that the term happens is restricted to the observation. We cannot describe what happens at least in the normal sense of the word between one observation and the next. In fact it's an illusion that the kind of existence the direct actuality of the world around us can be extrapolated into the atomic range. The quantum weirdness of the uncertainty principle and of our inability to measure more than one component of angular momentum are expressed formally in the idea of commuting and non-commuting operators. One reason we want to discuss this is that it's a central reason why it can be said that nobody truly understands quantum mechanics. In fact it'll come up directly in a future video when we investigate one of the strangest conclusions of quantum mechanics, a conclusion that calls into question our very conception of reality. This was brought to light in a 1935 paper by Einstein, Podolski and Rosen titled Can Quantum Mechanical Description of Physical Reality Be Considered Complete? Their argument rests on the fact that, quote, in quantum mechanics in the case of two physical quantities described by non-commuting operators the knowledge of one precludes the knowledge of the other. We say that multiplication is commutative. Two times three equals three times two. Equivalently, two times three minus three times two equals zero. Multiplication is commutative because it doesn't matter in which order we multiply two numbers. We get the same result in either case. On the other hand, division is not commutative. Two divided by three is not equal to three divided by two, or two-thirds minus three halves is not equal to zero. With division it does matter in which order we divide two numbers. Let's make a cartoonish illustration of this with the idea of commuting measurements. Suppose we have our cup of coffee again and we want to know how deep the coffee is. We stick a ruler in the coffee and measure eight centimeters. We also want to know the coffee's mass. So we put it on a scale and let's assume the scale is calibrated to read zero with an empty cup and we measure 200 grams. We want to check our depth measurement again so we put the ruler back into the cup and again measure eight centimeters. To make sure we got the correct mass we put the cup back on the scale and again we measure 200 grams. We can make these measurements as often as we want and in any order and we always get the same depth and mass. In this case we're justified in claiming that the coffee really has a depth of eight centimeters and really has a mass of 200 grams. Now suppose instead that after measuring a depth of eight centimeters and a mass of 200 grams we again measure the depth and obtain a result of 11 centimeters. Measure the mass again and find 140 grams. Our measurements are non commuting. In this case we simply have no physical basis for saying that coffee has a specific depth or mass. Suppose further that after measuring a depth of eight centimeters we remove our ruler. A while later we measure the depth and again find its eight centimeters. We repeat this any number of times and always obtain the result eight centimeters. Then we're justified in saying that the coffee is quote in reality eight centimeters deep. But as soon as we make a mass measurement subsequent depth measurements are no longer consistently eight centimeters. Then we're no longer justified in talking about the coffee's depth as something that exists quote in reality. In quantum mechanics observations are represented as operators. Two operators p hat and q hat commute. If p hat applied to q hat applied to the wave function equals q hat applied to p hat applied to the wave function. In other words if the order of making the observations doesn't matter. Conversely two operators are non commuting if the order does matter. It's convenient to define the commutator of two operators as the difference of these two orders of operation. And a zero commutator means the operators commute and a non zero commutator that they don't. In an appendix video we show that two operators that satisfy the uncertainty principle such as position and momentum have a commutator equal to the imaginary unit i times h bar. Now when we write delta x and delta p to represent the uncertainty and position and momentum we're being a bit imprecise about how we're defining uncertainty. A rigorous approach is to use the concept of a statistical standard deviation and in this case we can obtain the precise statement that the product of the standard deviations of position and momentum are rigorously greater than or equal to h bar over two which is h over four pi. Also in the appendix video we show that for the three components of an angular momentum operator j hat. The commutator of j hat x and j hat y equals i h bar j hat z and the other two similar relations shown. For the operator j hat squared representing the square of angular momentum magnitude we show that the only possible values we can measure are j times j plus one times h bar squared where j equals zero one half one three halves two and so on. That is a non negative integer zero one two etc as we saw for orbital angular momentum l or a positive half integer such as one half three halves and so on. The z component of angular momentum then has the form m sub j times h bar where the possible values of m sub j are minus j minus j plus one and so on up to j for a total of two j plus one possible values. This result will enable the development of the concept of quantum mechanical spin that we will cover in the next video.