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Quantum Jumps - The source of the problem with Bohr's Model
After identifying limitations in Bohr's model, the next wave of explanations of the nature of the atom arose. These included De Broglie's matter waves, but even these were limited.
The wave function
The debate between Albert Einstein and Niels Bohr about the electron and the atom became a personal conflict between Erwin Schrödinger on Einstein's side and Werner Heisenberg on Bohr's side. Schrödinger developed the theory of wave mechanics whereas Heisenberg developed Matrix Mechanics.
Both these theories are now accepted to be correct interpretations. The wave function is a useful way of trying to explain quantum phenomena.
In the following video clip Dr. Jim Al Khalili explain how the interference patterns of electrons in the two slit experiment can be explained using quantum mechanics and the idea of the wave function. Jim uses the analogy of a spinning coin to explain the concept of the wave function.
Dr. Al Khalili - Electron Interference
In 1926 the German physicist Max Born gave the wave function meaning in physical terms by proposing that,
"The square of the amplitude of the wave function at any point in space is directly proportional to the probability of locating the particle at that point."
This can be expressed as an equation as shown to the right.
The point in space, or unit volume, is represented by ΔV, the probability of locating the particle, or probability density, is represented by P (r). The square of the amplitude of the wave function is represented by |ψ|2.
Another analogy for the wave function was given by Dr. Jim Al Khalili in his book "Quantum: A guide for the perplexed". He describes the wave function to be like a crime wave!
He disappears and the police are attempting to locate him. Initially the houses in danger of being burgled are near the prison, however as time passes and this one man crime wave is still not caught, the area in which he might be found grows (the "position wavefunction" spreads out)
There is of course a higher percentage chance that he will be found nearer banks or rich peoples houses and less percentage chance of finding him in poorer areas or near churches and libraries. The crime wave, just like the wavefunction, is not a real thing but is simple a set of numbers assigned to positions on the map.
Suddenly the police receive a phone call from a house that has been burgled and the positon of the thief is now consigned to a small area around the house (the wavefunction "collapses" and the position is well known).
Before the police arrive however the thief slips away again and the crime wave spreads out once more.
The uncertainty principle
Watch the video clip below where Jim reveals how Heisenbergs frustration with Schroedingers wave equation led to the idea of the uncertainty principle.
Heisenberg's uncertainty principle can be expressed in the form of an equation where x represents the position of a particle and p represents the momentum.
As the certainty in the position of a particle increases the certainty in the momentum of the particle decreases. The product of these two values must be greater that h/4(pi)
The uncertainty principle can also be applied to other pairs of quantities like energy E and time t. This allows for the process of quantum tunneling, described below.
Watch the clip below where Jim visits Brussels and the site of the Solvay convention on the topic of electrons and photons in 1927, the greatest ever meeting of physics minds, where the future of atomic physics was mapped out.
Pair production and annihilation
The uncertainty principle can also explain how particle-antiparticle pairs can be produced out of energy and then annihilate each other. This gives new meaning to the idea of empty space.
Watch the video clip below where Jim explains that there is more going on in empty space than you might think.
Quantum tunnelling
The wave function can also be used to explain quantum tunnelling. How did Mr McNeill manage to break out of prison in the first place? While in prison, the wave function that describes our intrepid one man crime wave also has amplitudes outside the prison. Similar to the image below, Mr McNeill's wave function spreads beyond the wall. There is a higher probablity that he will be located inside the wall but there is still a small probability that he will be found outside the wall.
This means that there is a probability of Mr McNeill existing outside the prison.
Mr McNeill can briefly disobey the law of conservation of energy and borrow an amount of energy to escape as long he gives it back within a time that satisfies the uncertainty principle. This is an an analogy for alpha particle decay. The chances of an alpha particle being found outside a decaying nucleus depends on the ratio of the square of the wave function outside the nucleus to the the square of the wave function inside the nucleus. This links to the decay constant of that nucleus.
Here's a Minute Physics explanation of tunnelling.
And here is a short explanation of a scanning tunnelling microscope (STM).
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