Sunday, 27 February 2011

Wave-particle duality

Compton scattering: the phenomenon that when the incident photon hits a free electron, the free electron will be scattered and change its direction, with photon with lower frequency bounced away. This gives the evidence that photon can exist in matter form.
Recall the wave nature of light: diffraction (single slit) and interference (Young's double slit)
Particle nature of light: photoelectric effect and Compton scattering.
Einstein states that E=pc, where E is the energy of the photon, p is its momentum and c is the speed of light. Rewrite the equation: E = pc = pfλ = hf, we have the wavelength-momentum relation: p = h/λ. This describes the particle characteristic of photon.
In 1924, de Broglie claims that λ = h/p in his doctorial thesis called de Broglie relation. The corresponding λ is called De Broglie wavelength. This describes the wave nature of a matter which is revolutionary at the time, which awards him the Nobel Prize in Physics in 1929.
Students should note that though p = h/λ and λ = h/p is mathematically equivalent but they should be proved independently since these two equation describes different things. Of course, their validity has been no doubt after some experimental effort, but this is beyond the scope of this set of note.
No matter wave behavior is observed in daily life because their wavelength is far too small to be observed. (smaller than 10-35m). However electrons are proved to have wave nature in experiment: a fast-moving electron has wavelength about 10-10m while the interatomic separation is about the same. Interference pattern was observed when electron beam hits nickel foil and reflect with different angles. The graph of frequency of electrons VS deflected angle shows several maxima and minima. Very soon, Sir George Thomson (son of J.J.Thomson) created another experiment which hits the electron beam on a metal foil and the scattered electron beams interfere. Rings are shown on the fluorescent screen, which is similar to result by X-rays. The process that "electron shown on fluorescent screen" is a particle behavior, so this experiment actually shows the duality nature of electrons.
Double-slit experiment of electrons was also done to show the wave behavior of electrons. Some small molecules like C60's wave behavior have been experimentally verified too.
More about quantum theory
We say that a matter is described by a wavefunction. We do not know the state of a matter until it's observed. The state of a matter is finite and among those "allowed state". For example the allowed state of a piece of stationery can be "on the floor" or "on the desk", but never "between the desk and the floor". Then "between the desk and the floor" is not an allowed state. We never know the exact location of a matter because an observation only allows us to observe one of the allowed states. Cases with probability > 0 can be one of the allowed states.
Heisenberg's uncertainty principle
In classical view, the location of a matter is absolute and exact. Under a probability-position graph, we see only one spike which the matter is exactly there. However in modern physics the probability actually forms a wave instead of a spike only. Take interference of light as an example: When we allow exactly one photon to pass through the double-slit, interference between the waves still occur. We don't know the exact location of the photon, but we know that there's a large probability to find the photon in those maxima. The resulting probability-position graph forms a wave.
In semi-classical view, we observe the state of a matter by receiving reflected waves like visible light for illumination. But the Compton scattering shows that the electron's momentum has been changed since the photon collide with electron. The photon only gives the momentum or position of the electron before it changes its states. Therefore we never know them exactly. The uncertainty principle states that ΔxΔp ≥ h/4π, where Δx is the position uncertainty and Δp is the momentum uncertainty. Again we don't observe uncertainty in daily life because the uncertainty is far too small and is neglected.
Assume T=0K, according to classical theory: kT = mc2/3, (c is the r.m.s. speed), its speed is zero (which is exact), but it is impossible to have a infinitely exact value for its momentum. As a result the atoms, in fact, contain zero-point energy to allow its quantum vibration.
Quantum tunneling
In classical physics energy conserved, when the potential barrier is higher than the K.E. that the matter have, it can't pass through the barrier. However in quantum physics, there's a certain probability that a electron pass through the potential barrier without obtaining enough energy. Most of the matter wave is reflected back while a small part of them pass through. The probability (or proportion) of wave passed through is proportional to a-x where x is the length of barrier and a is a constant. i.e., is exponentially related.

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