It was difficult to make equations built for continuous spectra (what you see when you put the sun's light through a prism) fit spectra that just have a few peak frequencies between which there is nothing. Almost everything that had already been learned about light and energy had been done with large things like burning candles or suns, and those large objects all produce continuous spectra. Even though these ordinary-sized things were easy to do experiments with, it had still taken a long time to figure out the law (physics)laws that govern them. Now physicists were dealing with things too small to see, things that did not produce continuous spectra, and were trying to find a way to at least get clues from what they already knew that would help them find the laws of these small and gapped-out light sources.
The original equations dealt with a kind of vibrating body that would produce a wave, a little like the way a reed in an organ would produce a sound wave of a characteristic frequency. So there was motion back and forward (like the vibrating of a reed) and there was an emitted wave that could be graphed as a sine wave. Much of what had earlier been figured out about physics on the level had to do with electrons moving around nuclei. When a mass moves in an orbit, when it rotates around some kind of a hub, it has what is called "angular momentum." Angular momentum is the way that something like a merry-go-round will continue to rotate after people have stopped pushing it. The math used for phase calculations and angular momentum is complicated. On top of that, Heisenberg did not show all of his calculations in his 1925 paper, so even good mathematicians might have trouble filling out what he did not say.
Even though many physicists said they could not figure out the various math steps in Heisenberg's breakthrough paper, one recent article that tries to explain how Heisenberg got his result uses twenty math-filled pages. Even that article is not easy to understand. The math started with some really hard stuff and would eventually produce something relatively simply that is shown at the top of this article. Getting the simpler result was not easy, and we are not going to try to show the process of getting from an outdated picture of the universe to the new quantum physics. We need just enough detail to show that almost as soon as Heisenberg made his breakthrough a part of how the universe works that nobody had ever seen before came into view.
Heisenberg must have been very excited but also very tired when, late at night, he finally made his breakthrough and started proving to himself that it would work. Almost right away he noticed something strange, something that he thought was an annoying little problem that he could make go away somehow. But it turned out that this little nuisance was a big discovery.
Heisenberg had been working toward multiplying amplitudes by amplitudes, and now Heisenberg had a good way to express amplitude using his new equation. Naturally he was thinking about multiplication, and about how he would multiply things that were given in terms of complicated equations.
Heisenberg realized that besides squaring amplitude he would eventually want to multiply position by momentum, or multiply energy by time, and it looked like it would make a difference if he turned the order around in these new cases. Heisenberg did not think it should matter if one multiplied position by momentum or if one multiplied momentum by position. If they had been just simple numbers there would have been no problem. But they were both complicated equations, and how you got the numbers to plug into the equations turned out to be different depending on which way you got started. In nature you had to measure position and then measure momentum, or else you had to measure momentum and then measure position, and in math the same general situation prevailed. (See the English Wikipedia article if you want to learn the fussy details!) The tiny but pesky differences between results were going to remain, not matter how much Heisenberg wished they would go away.
At the time Heisenberg could not get rid of that one little problem, but he was exhausted, so he handed his work in to his immediate supervisor, Max Born, and went on vacation.
Max Born was a remarkable mathematician who soon saw that the equation that Heisenberg had given him was a sort of recipe for writing a matrix. Dr. Born was one of the few people at that time who was interested in this odd kind of math that most people figured was not good for very much. He knew that matrices could be multiplied, so doing all the calculations for accounting for one physics problem could be handled by multiplying one matrix by another. Just being able to put a complicated procedure into a standard and acceptable form would make it easier to work with. It might also make it easier for other people to accept.
Born was such a good mathematician that he almost immediately realized that switching the order of multiplying the two matrices would produce a different result, and the results would differ by a small amount. That amount would be h/2πi. In everyday life, that difference would be so small that we could not even see it.
It took a couple of years, but Heisenberg was able to prove the Uncertainty Principle, which says that Δx × Δp = h/2, which is the number that comes out of the original equations but leaves out the π and the i that have to do with phase changes. Heisenberg explained that he derived his uncertainty principle from this earlier result when he wrote a paper in 1927 introducing this theory.
The constant written h, called the Planck constant, is a mysterious number that often occurs, so we need to understand what this tiny number is. Numerically, it is usually given as 6.62607×10^-34 J s (joule seconds). So it is a quantity that involves energy and time.
It was discovered when Planck realized that the energy of a perfect radiator (called a black-body radiator) is emitted in units of definite size called "quanta" (the singular of this word is "quantum"). Radiated energy is emitted as photons, and the frequency of a photon is proportional to the "punch" it delivers. We experience different frequencies of visible light as different colors. At the violet end of the spectrum, each photon has a relatively large amount of energy; at the red end of the spectrum each photon has a relatively small amount of energy. The way to calculate the amount of energy of a photon is given by the equation E = hν (energy equals the Planck constant times "nu" or frequency).
The Heisenberg uncertainty principle Δx × Δp ≥ h tells us that whenever we try to pin down certain pairs of numbers we can only get so close, and that if we try to get clearer on one of them, i.e., if we try to make Δx smaller so we have a better idea of the position of something, then we will have to get handed back a bigger number for the other number of the pair, and that the amount the two are off by is closely related to h.
Another pair of physical quantities goes according to the uncertainty relationship: ΔE × Δt ≥ h, and that pair indicates, among other things, that if we look in interstellar space, some place where we would not expect to find anything at all, and we reduce Δt closer and closer to 0, then to keep the balance shown in the equation ΔE has to get larger and larger — and suddenly something with momentum can pop into existence just for that brief period of time.