In classical physics matter is made from tiny particles and electromagnetic radiation is made from spread out waves. In quantum mechanics matter can behave like waves and radiation can behave like particles. This is called the wave-particle duality. Quantum mechanics is necessary to describe the behavior of matter on the scale of atoms and below, and it is also important at large scales for special systems like metals, superconductors, white dwarf and neutron stars, and maybe our brains.
The late Richard Feynman said that the two-slit experiment is "the only mystery" both to the novice and the experienced physicist because it is so alien to our ordinary outer experience described so well by classical mechanics .
One can do a two-slit experiment with an electron beam. The electrons are always detected in tiny lumps like particles, but the probability distribution of their detection is a continuous wave pattern showing interference fringes as long as there is no way to measure which slit the electron goes through. If there is such a way the fringes disappear. One can do the experiment at very low particle flux so that there is only one electron in the apparatus at a time. Wait long enough and one sees the wave fringe pattern build up. Therefore, we are forced by the facts to conclude that in some sense and electron is interfering with itself. This is the central mystery that Feynman is describing. How can a tiny particle be at both slits? One way to picture this is to say that the electron passes through only one slit in two parallel universes [Go Live] that then interfere with each other to make our universe. Some physicists think this is the right way to think about the mystery. In opposition to that, David Bohm is able to explain the mystery in terms of the mathematically precise, but "spooky", quantum force from both slits guiding each electron which really only goes through one slit. In contrast, the pragmatic Copenhagen interpretation of Niels Bohr makes no attempt to imagine actual tiny electrons at all.
Feynman in his Cal Tech lectures uses the Copenhagen interpretation. He has a global point of view in that his basic tool is a probability amplitude for a process through space and time. The amplitude is a complex number that can be pictured as an arrow in a plane. The squared length of this vector is proportional to the probability that the process will be measured. The orientation of this vector in the plane is the quantum mechanical "phase". The absolute phase has no physical meaning because the choice of coordinate system is arbitrary. Only the difference in phases, (i.e., the angles between pairs of vectors) is invariant (i.e., does not change) under a rotation of the coordinate system. Therefore, only the relative phase has physical meaning. Indeed, the fringe pattern in the two-slit experiment can be explained as the variation in the relative phase of the two amplitudes, each describing the path of the single electron from a slit to a given point on the screen behind the slits.
The first equation of wave-particle duality was discovered by Max Planck in 1900 when he successfully explained the distribution of energy E with frequency f in black body radiation that is in thermodynamic equilibrium with the atoms in the walls of a container. Classical physics was in crisis because it predicted that the intensity of radiation would explode to infinity with increasing frequency. Experiment showed a nice decrease of the intensity with increasing frequency. Planck made a wild guess that the transfer of energy between atoms and radiation at a fixed frequency f was not continuous but happened in integer multiples of that frequency. The constant of proportionality is h and is called Planck's constant. It is very tiny equal to about 6.6*10^-34 Joule-seconds. Planck's formula was
Einstein extended Planck's idea five years later to explain how light ejects electrons from metals. The energy of these electrons only depended on the frequency and not the intensity of the radiation falling on the metal. If the frequency was too low no electrons would be ejected no matter how intense the radiation. The number of electrons ejected did depend on the intensity. Einstein had the second important wild idea that light waves really consisted of "photons" each of energy E = hf. E is a particle property while f is a wave property. We know from the mathematics of Fourier analysis that a wave of single frequency f must extend infinitely in time. The analogous oscillations of the wave in space in a given direction are described by a wave number k. For radiation f = ck. A wave of definite k must be a periodic pattern extending infinitely along the direction or propagation in space. The distance between successive crests is the wavelength 2pi/k = lambda. So how is the tiny particle related to the spread out wave?
Louis DeBroglie, shortly after WWI, had a third wild idea in which he applied the wave particle duality and special relativity to matter. He predicted that particles like electrons could also behave like waves. A particle of momentum p in a given direction would have a wave length lambda such that p = h/lambda. In fact the wavelike diffraction of electron beams in crystals was observed a few years after DeBroglie predicted it.
Schrodinger built upon De Broglie's crazy idea to develop a wave equation which explained atomic spectra much better than Bohr's atomic mechanics which implicitly had DeBroglie's idea in it before DeBroglie rediscovered it. The important point is that Bohr and others did not see the physical implications of a certain formal step in the atomic mechanics of electrons bound in atoms to their free behavior in beams. The new mechanics was so alien to classical physics that the great geniuses were mind-boggled and only slowly realized what they were doing. In fact, as Feynman points out we still don't really adequately understand quantum mechanics and its limits and how it makes the apparently classical world even though we can make astoundingly accurate computations of some experimental numbers using its mathematics.
Heisenberg formulated the uncertainty principle. Feynman emphasizes that quantum mechanics really depends upon this principle never being violated. Yet, David Albert, in his book, The Quantum Mechanics of Experience, has recently argued that, under very special conditions, a new kind of quantum system that measures itself can violate Heisenberg's principle obeying the strict rules of quantum mechanics. So we have a dilemma to explore. Does quantum mechanics destroy itself? Is it an inconsistent theory? Or, has David Albert made a mistake? Or, have I misunderstood what he said? This is a homework problem for later on in the course! :-)
http://www.hia.com/hia/pcr/f.html
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