We may bring that experience to the world of atomic-sized phenomena and incorrectly assume that if we measure the position of something like an electron as it moves along its trajectory it will continue to move along that same trajectory, which we imagine we can then accurately detect in the next few moments. We need to learn that the electron did not have a definite position before we located it, and that it also did not have a definite momentum before we measured the trajectory. Moreover, we may justifiably assume that a photon produced by a laser aimed at a detection screen will hit very near to its target on that screen, and confirm this prediction by any number of experiments. Next we will discover that the more closely we try to pin down some location for the electron on its way toward the detection screen, the more it and all others like it will be likely to miss that target. So pinning down a location for an electron makes the trajectory more indefinite, indeterminate, or uncertain. If the trajectory were made more clear and then we were to try to locate that electron along an extension of the trajectory we just staked out, then we would find that the more precise we made our knowledge of the trajectory, the less likely we would be to find the electron where ordinary expectations would lead us to believe it to be. If pitchers threw electrons instead of baseballs, and an overhead camera and side-facing camera were placed somewhere between the pitcher's mound and home plate so that the exact position of the electron could be determined in mid flight, then without the cameras being turned on, the pitcher would throw straight balls, and with the cameras turned on his pitches would start out straight but gyrate wildly after their pictures were taken. The more clearly we know where the ball was halfway toward home place, the more trouble the batter will have in getting ready to hit it with his bat.
Unexpected consequences of the uncertainty feature of nature support our understanding of such things as nuclear fission, the control of which gave humans a new and very powerful source of energy, and quantum tunneling, which is an operating principle of the semiconductors that are so important to modern computer and other technologies.
In technical discussions one almost always talks about position and momentum. Momentum is the product of velocity and mass, and in physics the idea of velocity is the speed that something is going in a certain direction. So sometimes one can also talk about the velocity of the thing in question and ignore its mass, and sometimes it is easier to understand things if we talk about the trajectory or path that something follows. That idea also includes the ideas of speed and direction. In the following diagrams we will show the main features of uncertainty in concrete terms, in the world of real things. Later we will use a little math to be able to give a clear idea of how much wiggle room there is between position and momentum.
Very shortly after Werner Heisenberg created the new quantum physics something unexpected came right out of his mathematics, the expression:<br>
These symbols put into math form what you have already seen in the pictures above. The symbols say, in a clear way, that you cannot be perfectly certain about where something is and where it is going. If you get clearer on where it is at any time then you have less of an idea on where it is going and how fast. If you get clearer on where it is going and how fast at any time, then you have less of an idea of where it is right now.
Scientists had already learned why certain substances give off characteristic colors of light when they are heated or otherwise excited. Heisenberg was trying to explain why these colors each have a characteristic brightness. It would not have been good enough if he and the other scientists had just said, "Well, that's just the way it is." They were sure that there had to be a good reason for these differences, and for the fact that the ratios among the bright line strengths were always the same for each sample of an element.
He had no idea that he was going to stumble over a hidden secret of nature when he set off to discover the explanation for the intensities of the colored lines characteristic of each of the elements. The study of quantum mechanics had already shown why hydrogen has four bright lines in the part of the spectrum that humans can see. It must have seemed that the next thing to learn would simply be how to calculate their brightness. Hydrogen seemed to be the obvious place to start since hydrogen has only one electron to deal with, and only four lines in the visible part of the spectrum. Surely there must be a good reason for their not being equally bright. The explanation for the brightness of the different-colored lines of neon and the other elements could wait.
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Heisenberg started working on quantum physics by adapting the classical equations for electricity, which are very complicated to begin with, so the math behind his 1925 paper was very hard to follow.
He was trying to find the right way to calculate the intensity of bright lines in the hydrogen lamp spectrum. He had to find a related quantity called "amplitude" and multiply amplitude by amplitude (or in other words he had to square the amplitude) to get the intensity he wanted. He had to figure out how to express amplitude in a way that took account of the fact that hydrogen lamps do not radiate at all frequencies, and do not radiate across a continuous range of frequencies in the part of the spectrum that people can see. Heisenberg found a remarkable new way of calculating amplitude.
The strange equation|equation that Heisenberg discovered and used to do the multiplication of one quantum quantity (e.g., position) by another (e.g., momentum) was published in what has been called "Heisenberg's 'magical' paper of July 1925." <br>
The math above looks very hard, but the math leading up to it is very much harder and is extremely hard to understand. It is given here just to show what it looked like. Heisenberg's paper is a historical landmark. Many of the physicists who read his paper said that they could not disagree with his conclusions, but that they could not follow his explanation of how he got to those conclusions. The beginning equations that Heisenberg used involved Fourier series, and involved many factors. We will come back to the equation above because it is a kind of recipe for writing out and multiplying matrices.
The new equations had to be so strange and unusual because Heisenberg was describing a strange world in which some things, such as the orbits of electrons, do not slowly get larger or smaller. The new kinds of changes involve jumps and large gaps between jumps. Electrons can only jump between certain orbits, and the energy gained or lost in changing between orbits is produced when a photon of the right energy is absorbed or a new photon of the right energy is produced. If electrons in hydrogen atoms most frequently jump down (fall) between two particular orbits, then more photons will be emitted at that energy level, and so the light produced at that level will be the most intense.