 So, the uncertainty principle expresses fundamental limits on our ability to predict certain outcomes of experiments with quantum particles, of which so far we've only looked at one, the photon. Suppose someone claims his laser will emit a photon of exact energy E at exact time t. The photon will travel at the speed of light to detect or distance L away, therefore the detector will absorb a photon of energy E at time t plus L over c. You could reject those claims out of hand because we know that it's not possible to speak of the energy E except within an uncertainty delta E and the time within an uncertainty delta t, where delta E times delta t is no smaller than Planck's constant. However, this only applies to making predictions. When the photon is absorbed by the detector, it will have arrived at a specific time t and it will have delivered a specific energy E. The uncertainty principle doesn't mean that the photon will deliver a range of energies, whatever that would mean. It tells us our limitations in being able to predict what that energy will be. After the photon is absorbed by the detector, it ceases to exist and we can no longer talk about the energy of the photon in the present tense, only in retrospect. Still, this idea of going from uncertainty before an experimental outcome to certainty afterward may seem a bit weird. As a rough analogy, consider the Galton board again. We're going to drop a ball through this maze of bumpers and it will end up in one of the bottom bins. Suppose the bumpers are hidden and I ask you to tell me where the ball will end up. You can't know for sure. The best you can do is to tell me the probabilities that it will end up in the various bins. Let's crudely call this probability our wave function. It expresses our uncertainty in the exact outcome of the experiment. While the ball is moving, hidden, through the bumper maze, this wave function continues to represent everything we can say about the final outcome. But when the ball finally comes out of the maze, it will end up in one and only one bin. At that moment, there is no longer any uncertainty. The ball is in that bin with 100% certainty. You might say that our wave function has, quote, collapsed to a single bin. Likewise, when a photon is traveling through the double slit experiment, everything we can say about its final position is represented by a wave interference pattern spread across the detector. But when it's detected, it ends up at a single pixel. The wave function has collapsed. We'll come back to wave function collapse in a future video. For now, let's go back to the uncertainty principle. The uncertainty principle was a topic of great contention in the early days of quantum theory. Einstein was among the most outspoken of those who really disliked the idea that there were fundamental limits on our ability to make physical predictions. He posed a series of challenges regarding the uncertainty principle. Niels Bohr, whose immense contributions to quantum theory we'll discuss in a future video, took up the challenges. Bohr was in the camp who believed that the uncertainty principle was a fundamental fact of nature. Einstein's challenges took the form of thought experiments. These weren't meant to be suggestions for practical experiments, but only experiments that were possible in principle without regard to any technological limitations. One of the most interesting whenists follows. Imagine Einstein says, you have a box sitting on a scale. The box has a hole in it. The hole is covered by a shutter mechanism driven by a timer. Inside the box is a photon. The inside of the box is perfectly reflective, so the photon just bounces around indefinitely. Because energy equals mass times the speed of light squared, the mass measurement we read off the scale will contain a contribution from the energy of the photon. Suppose the scale reads a mass of M1. In principle, this could be determined with arbitrary precision. At a predetermined time, the shutter mechanism quickly opens and closes. In principle, the time the shutter is open can be made arbitrarily small. Suppose the photon happens to exit the box during this time. If we now measure the box's mass, we will necessarily find a smaller value, called this M2. This can also, in principle, be measured with arbitrary precision. The difference between M1c2 and M2c2 is the energy of the photon. So we would be able to say that the box emitted a photon at time t with arbitrarily small uncertainty delta t, and with energy e with arbitrarily small uncertainty delta e. And notice that we didn't need to destroy the photon during this experiment, so those arbitrarily precise statements could be used to predict future events. But the uncertainty principle says that the product delta e times delta t can never be less than Planck's constant. Therefore, Einstein argued, the uncertainty principle cannot represent a fundamental limit of nature. This was a difficult challenge, but more eventually found a convincing response. A subtle issue arises in the process of weighing the box. If a scale pointer indicates a mass M, we assume this means that the downward force of gravity acting on the mass, M times the gravitational acceleration g, is exactly balanced by the upward force of the scale's spring. But what if I claim that, no, the box mass is actually a little more, M plus delta M? If this is true, then there is a net downward force, delta M times g, acting on the box. To choose between these claims, you have to show that this net force does or does not exist. How can you do this? A force produces an acceleration, an acceleration acting over a given time produces a given change in momentum. So if this net force is present, and we watch the box for a time t, we'll see it gain a momentum delta P equal to delta M times g times t. That is the box will start to fall. Turning this around, if delta P is the precision with which we can measure momentum, then the precision with which we can measure mass is delta M equals delta P over g times t. T is the length of time we watch the box looking for signs of acceleration. Bohr's point is that to measure the box's mass we not only have to see where the scale arrow is pointing to, but we also have to verify that the arrow stays put. For example, from this picture can you conclude that the person standing on the bath scale weighs 100 pounds? Not really. Maybe the person just stepped on the scale. She weighs 110 pounds and the arrow, which started at zero, is still moving towards its final position. Or maybe she weighs 90 pounds and the arrow overshot up to 100 and is moving back towards 90. You'd have to view the scale over some time interval to conclude that the arrow points to 100 and is not going to move. So far we've discussed the uncertainty principle applied to photons. Later we'll see that it's a general principle that applies to any particle or object. Applying it to the box we can say that the minimum uncertainty we can have in the box's momentum equals Planck's constant over the minimum uncertainty we can have in the box's position, delta X. Substituting this into the previous equation, we have that the minimum uncertainty with which we can know the box's mass, delta M, equals Planck's constant over G times T times delta X. Now Einstein is hoist upon his own patard. Bohr invokes the general theory of relativity. According to relativity, clocks at different heights in a gravitational field run at different rates. If one clock is at X equals 0 and a second is at X equals delta X and the rate of the first is taken to be 1, then the second runs at a rate 1 plus G delta X over C squared. This is gravitational time dilation, link below to the video in the relativity series where this is discussed. Due to their different rates, after running for a time T, the clocks will differ by delta T equals G delta X over C squared times T. We rewrite this expression as C squared delta T equals G T delta X. Previously we established delta M equals H over G T delta X. We rewrite this as G T delta X equals H over delta M. Now two expressions are equal to G T delta X, so they're equal to each other. C squared delta T equals H over delta M. We write this as delta M C squared equals H. Finally, because E equals M C squared, delta M C squared is delta E, and we have the uncertainty principle. Uncertainty in energy times uncertainty in time equals Planck's constant. To this day, the uncertainty principle has resisted any and all attacks. The overwhelming consensus based on over a century of thought, research and experiment is that there are indeed fundamental limits to the precision with which we can predict the outcome of certain phenomena.