Tutorial PHYS1002 Sheet No. 7
Study guide
Do not confuse the probability density with probability. The probability density is related to probability by P(x)dx = probability. Remember Bohr's correspondence principle: In the limit of very large quantum numbers, the classical calculation, and the quantum calculation must yield the same result. The expectation value of x gives the average value of x, which one obtains if a large number of identical experiments is carried out. For a particle in a box, well, etc, the smallest possible value of energy is not zero as in the classical theory. This follows from the uncertainty relation.
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Problems
1) The Compton effect
X-rays of wavelength 22 pm (photon energy = 56 keV) are scattered from a carbon target, the scattered radiation being viewed at 85 to the incident beam. a) What is the Compton wavelength shift? b) What percentage of its initial energy does an x-ray photon lose?
2) De Broglie wavelength
An electron is in a narrow molecule 6.5nm long, a situation that approximates a one-dimensional infinite square well. If the electron is in its ground state what is the maximum wavelength of electromagnetic radiation that can cause a transition to the excited state?
3) Expectation values, probabilities and energies
A particle is confined to a one-dimensional box of length l a) Calculate the expectation value x of the position of the particle inside the box. b) The particle is in the first excited state (n=2). Calculate the probability of finding the particle within some small distance dx of x = l / 2 . c) Calculate the most probable positions inside the box at which one can find the particle being in the first excited state (n=2). d) Four more particles are added to the box. What is the ground-state energy of all five particles if they are bosons? What would be the ground state energy of all the particles if they were fermions? 2 n x e) Wave function of the particle in the box is ( x ) = sin . l l HINT: sin 2 = 1 (1 cos 2 ) 2
4) Energies, transitions and expectation values in a one dimensional well
An organic molecule has a chain of several carbon atoms along a line. An electron in the molecule behaves as if confined to a one-dimensional box of length 1.55nm. a) Determine the energies of the 5 lowest states. b) Assuming that transitions can occur between any of these 5 levels, which of these transitions will produce visible light (400nm-700nm)? c) For the first excited state (n=2), find the most probable positions of the in electron the molecule.
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5)
This question is for discussion in the tutorial. Rather than finding an answer, the aim is to explicitly work out how to approach the problem: For an electron confined in a one-dimensional box of length L, determine which transitions can produce/absorb visible light.
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Mad Physicist's Section
Photonic crystals boost semiconductor lasers
30 October 2003 PhysicsWeb (http://www.physicsweb.org/)
Physicists have made a new type of laser by combining a quantum cascade laser with a photonic crystal. Raffaele Colombelli of Bell Labs in the US and colleagues say that their novel proof-of-concept device could find use in sensing applications and fundamental research in optics (R Colombelli et al. 2003 Sciencexpress 1090561). Conventional semiconductor lasers emit photons when electrons in the conduction band and holes in the electron band recombine. The wavelength or energy of the photon is determined by the energy difference between the conduction and valance bands - which is a fundamental property of the semiconductor. Quantum cascade lasers, on the other hand, emit light when electrons fall from a higher to a lower energy level in a quantum well. Since the quantum well can contain a whole series of levels, the same electron can emit a large number of photons as it cascades down through the levels. Moreover, the wavelength depends on the width of the quantum well which means that photons of more than one wavelength can be emitted by the same device. Howev...