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    Describe the experimental evidence for wave-particle duality. State the uncertainty and correspondence principle.

    © BrainMass Inc. brainmass.com December 24, 2021, 9:53 pm ad1c9bdddf

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    WAVE-PARTICLE DUALITY & the Experimental Observations to back it up:
    Wave-particle duality states that all matter and energy exhibit wave-like and particle-like behaviour. This can be seen in several experiments.

    Firstly Young's slit experiment, which involves sending collimated light through a double slit, with widths of the order of magnitude of the wavelength of the light, and measuring the interference pattern on a screen a short distance away. The intensity on the screen varies as cos2, meaning that at some angle the light from both slits is in phase and constructively interfering to form maxima, and that at other angles it is completely out of phase and destructively interferes to give intensity minima. This has been carried out with light of very low intensities so that you can see one photon at a time hit the screen, but the same interference pattern builds up over time. This suggests that each photon is acting like a wave. The experiment has also been done with neutrons and heavier particles and the same pattern results; again suggesting that these particles are acting as waves.

    Secondly electrons fired at a crystal are seen to diffract in certain directions depending on their incident energy. This was first observed by Davisson and Germer using a Nickel crystal target in 1927. A heated filament acts as an electron gun and the electrons are accelerated towards the crystal at normal incidence. A detector at a certain angle measures the number diffracted. If the electrons act like waves, then in certain directions they will superimpose constructively giving peaks and in other directions they will superimpose destructively giving minima. The position of these peaks is determined by Bragg's law: nl=2dsinx where l is the wavelength, n the order of the peak, d the separation of atoms and x the angle of incidence, which means we can measure the effective wavelength of the electrons; roughly 0.17nm. This shows that the position of the peaks depends on wavelength and therefore by de Broglie's relation on incident energy. Again this also has been carried out with neutrons, alpha-particles, helium atoms etc, and the effect is the same. This suggests the particles are acting like waves.

    Another phenomenon which supports wave-particle duality is quantum mechanical tunneling. A particle in a potential well is not classically expected outside the well because it doesn't have enough energy to overcome the potential barrier. Quantum mechanically there is some finite probability of finding the particle outside the well because Schrödinger's equation gives solutions of the form f= exp(-kx)*exp(iwt). This is a decaying wave known as an evanescent wave, and because the probability is given by |f*f| there is some chance of finding it outside the well. This has been observed in alpha decay of radioactive nuclei; the nucleus acts as a potential well and there is a very strong relationship between the half-life of decay and the energy released, which can only be explained by quantum mechanical tunneling. It is also used in Scanning Tunneling Microscopes. These work by keeping the tunneling current, between a sample and atomically fine metal tip with some voltage difference, fixed whilst scanning across the surface using piezoelectric crystals to alter the height of the tip. As the tunneling current is a strong function of distance, the tip must be kept at the same distance from the surface and hence we can tell the surface relief of the sample. Therefore we must consider particles as probability waves.

    These three examples give evidence for de Broglie's hypothesis that the wavelength of a particle is given by h/p where h is the Plank constant and p the momentum, and this wavelength matches that found experimentally. If the particle is behaving like a probability wave there is some uncertainty as to where it is located. This leads to the idea of the Heisenberg uncertainty principle which states that the uncertainty in position Dx and the uncertainty in momentum Dp are restricted by DxDp>=ħ/2. If we assume the particle has a Gaussian wavepacket then its uncertainty in position is just the standard deviation s. If we then Fourier transform this into k-space the uncertainty in k is given by 1/s. As momentum p= ħk we obtain DxDp=ħ. A fuller treatment gives the uncertainty relation.

    Finally it is obvious that these effects of wave-particle duality are only seen at small scales, and the classical picture works adequately in the limit of large numbers of particles. This is known as the Correspondence Principal, and was first stated by Bohr.

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