Pedestrian Notes on Quantum Mechanics

According to Kelvin a collection of thoughts cannot advance to the stage of Science without numbers. Any observable of interest in physics should be measurable or expressed in terms of measurable quantities. Length and time are two of the indefinables of classical mechanics, since on an intuitive base there are no simpler or more fundamental quantities in terms of which length and time may be expressed. The problem of space-time picture of the physical world is connected with the rigour of exact description of nature requiring, say, differential equations, by means of which we are able to gauge the intuitive space-time scales of any motion. The full number of indefinables in mechanics is three, as all its quantities could be expressed by only three indefinables. The third mechanical indefinable is usually the mass, but also force may be chosen [1]. Human beings in their everyday lives are continuously “measuring” the mechanical indefinables, as well as other indefinables of physics, e.g., the temperature, by means of their physiological senses. Of course, it is a very rough “measurement”, because it could be expressed in words, not in numbers. Words and numbers are complementary units of knowledge. Pure numbers do not tell us much about Nature unless we assign them some significance. As a good example consider the number 3.52. Just a (real) number as any other. But now write it as 2π/e . For some physicists it has already a meaning. Finally, let us write down the full chain, i.e., 2∆0/Tc = 2π/e γ = 3.52. It tells us that 3.52 is the BCS value for the ratio between the gap at zero temperature and the critical temperature for the transition to the superconductor phase. One gets 3.52 only in BCS theory. Because of its beauty, I am temted to give a second example which has been quoted by Noyes [2] from the books of Stillman Drake on Galileo. So, what about the number 1.1107... Nothing special at first glimpse. But now let us give a first significate: 1.1107... = π/2 √ 2. Geometrically it is the ratio between the quarterperimeter of the circle and the side of the square inscribed into that circle. Geometrical (i.e., spatial) measurements and thinking were much developed by Old Greeks. But Galileo was the first to give 1.1107... as the ratio of two times, namely the time tp it takes for a pendulum of a specific length l to swing to the vertical through a small arc to the time tf it takes a body to fall from rest through a distance equal to the length of the pendulum. Galileo’s measurement was 942/850 = 1.108, but he was not aware he measured π/2 √ 2. However he gave a remarkable formulation of the “law of gravity”. The Galilean gravitation states that the ratio of the pendulum time to the falling time as specified above is the constant 1.108, “anywhere that bodies fall and pendulums oscillate”. To obtain the number of indefinables (NOI) a community of physicists should adopt rules of their measurements, especially since NOI is not a fixed number, and new types of experiments might augment NOI. The rule of classical indefinables is to choose a durable standard of unit for each of them and to have a good dividing engine. This has been achieved rather easy for the meter of length but not so easy in the case of the meter of time (second). In the latter case the great difficulty has been for a long time the missing of an accurate dividing machine. Large errors were continuously accumulating