Heisenberg's idea is that if something like an electron were shot into a big box with a certain speed and in a certain direction, then it would be possible to calculate a fairly accurate idea of what its path would be like in the future. However, if the box were made smaller then we would have a more certain idea of where it was located, and because of that we would have to have a less certain idea of how it was traveling. American physicist Brian Greene gave the analogy of a moth which flies placidly around in a large closet but who flies frantically back and forth and up and down when placed in a glass jar. Another interesting uncertainty phenomenon, which makes possible many electronic devices, is called quantum tunneling. In our everyday life, people cannot move through walls. However, electrons can move through solid walls. In the animation to the right, you can see a faint white puff on the right side of the wall after a big puff hits the wall from the left. That dim spot of light represents the photon or other atomic particle which tunnels through the wall.
Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect. This says that measurements of some systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level as a physical "explanation" of quantum uncertainty.
However, it is now clear that the uncertainty principle is a property of all wave-like systems. It arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, "the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology". "Measurement" does not mean just a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer</u/l>.
The uncertainty principle came from Werner Heisenberg's matrix mechanics. Max Planck already knew that the energy of a unit of light is proportional to the frequency of that unit of light (formula_2), and that its amount of energy can be expressed in familiar terms such as the joule by using a proportionality constant. The constant he gave the world is now called the Planck constant and is represented by the letter h. When matrices are used to express quantum mechanics, frequently two matrices have to be multiplied to get a third matrix that gives the answer the physicist is trying to find. But multiplying a matrix such as P (for momentum) by a matrix such as X (for position) gives a different answer matrix from the one you get when you multiply X by P. The result of multiplying P by X and X by P and then comparing them always involves the Planck constant as a factor. The number used to write the Planck constant will always depend on the system of measurement in use. (With a certain system of measurement, its numerical value is one.) The slope of the line in the diagram to the right that shows the ratio of frequency to energy will also depend on the system of measurement chosen.
The following diagrams show what happens when we try to measure both location and momentum.
The practical result of this mathematical discovery is that when a physicist makes position more clear then momentum becomes less clear, and that when the physicist makes momentum more clear then position becomes less clear. Heisenberg said that things are "indeterminate," and other people liked to say that they were "uncertain." But the mathematics shows that it is the things in the world that are indeterminate or "fuzzy," and not that it is just that humans are uncertain about what is going on.
Here we will show the first equation that gave the basic idea later shown in Heisenberg's uncertainty principle.
Heisenberg's groundbreaking paper of 1925 does not use and does not even mention matrices. Heisenberg's great success was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)" of hydrogen radiation.
After Heisenberg wrote his breakthrough paper, he gave it to one of his teachers to fix up and went on a vacation. Max Born was puzzled by the equations and the non-commuting equations that even Heisenberg thought were a problem. After several days Born realized that these equations were directions for writing out matrices. Matrices were new and strange, even for mathematicians of that time, but how to do math with them was already clearly known. He and a few others worked everything out in matrix form before Heisenberg came back from his time off, and within a few months the new quantum mechanics in matrix form gave them the basis for another paper.
Max Born saw that when the matrices that represent pq and qp were calculated they would not be equal. Heisenberg had already seen the same thing in terms of his original way of writing things out, and Heisenberg may have guessed what was almost immediately obvious to Born—that the difference between the answer matrices for pq and for qp would always involve two factors that came out of Heisenberg's original math: Planck's constant h and i, which is the square root of negative one. So the very idea of what Heisenberg preferred to call the "indeterminacy principle" (usually known as the uncertainty principle) was hiding in Heisenberg's original equations.
Heisenberg had been looking at changes that happen in an atom when an electron changes its energy level and so gets closer to the center of its atom or gets farther from its center, and, especially, situations in which an electron falls to a lower energy state in two steps. Max Born explained how he took Heisenberg's strange "recipe" for finding the product, C, of some change in an atom from energy level n to energy level n-b, which involved taking the sum of multiplying one change in something called A (which could be, for instance, the frequency of some photon) produced by a change of energy of an electron in the atom between energy state n and energy state n-a) by a succeeding change in something called B (which could be, for instance, the amplitude of a change) produced by another change in energy state from n-a to n-b):
and discovered something ground breaking:
By consideration of ...examples...[Heisenberg] found this rule... This was in the summer of 1925. Heisenberg...took leave of absence...and handed over his paper to me for publication...
Heisenberg's rule of multiplication left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory...Such quadratic arrays are quite familiar to mathematicians and are called matrices, in association with a definite rule of multiplication. I applied this rule to Heisenberg's quantum condition and found that it agreed for the diagonal elements. It was easy to guess what the remaining elements must be, namely, null; and immediately there stood before me the strange formula
Later on, Heisenberg put his discovery into another mathematical form: