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Problem Set #7 MAT204 due 29.04.2010 1) An electron is trapped in a one-dimensional energy well of width L with rigid walls. (a) For L = 1 nm,

Materials Science course homework which is about properties of electrons and their movements in advance.
MAT204 Problem Set #7 due 29.04.2010 1) An electron is trapped in a one-dimensional energy well of width L with rigid walls. (a) For L = 1 nm, calculate the wavelengths of the three lowest-energy photons capable of exciting electrons from the ground state. (b) At what value of L will the energy gap between the two lowest energy levels equal k B T at T = 300 K? (c) For L = 1 cm, what would be the quantum number of the energy level k B T above the ground state at T = 300 K? 2) A 10 eV electron is incident upon a step potential barrier of 1 2 eV. By how much will the wave function and the probability density of the electron decrease (a) after penetrating 0.1 nm into the barrier? (b) after penetrating 1 nm? 3) Nine electrons are injected into a one-dimensional energy well with a width of 2 nm. (a) What will be the quantum numbers and energies of each of the nine electrons? (b) What is the lowest-energy photon that can now be absorbed by electrons in this energy well? 4) Electrons are incident upon a 5 eV step energy barrier. Calculate the reflectivity of the barrier (the ratio of the probability density of the reflected electron wave to that of the incident electron wave) for 10 eV and for 20 eV electrons. How does this result differ from what would be expected classically? 5) Calculate the possible energy levels of an electron in a one-dimensional energy well 1 nm in width and 10 eV in depth. (Because the well is finite in depth, the wave function is nonzero outside the box, the simple solution for the well with rigid walls no longer applies, and you must use the boundary conditions described in the text. For simplicity, consider only symmetric wave function. You will end up with an equation including a trigonometric function, which can be solved graphically or numerically. You should end up with only three energy levels, although there are also two allowed energy levels associated with antisymmetric wave function.) Compare the resulting energy levels with the corresponding energy levels of a well of the same width with rigid walls. 6) What percentage of the full nuclear charge is “screened” by other electrons for the 1 s , 2 s , 2 p , and 3 s electrons of sodium? 7) What is the most probable radius of 1 s electrons in He + and Li +2 ? Prove these values. 8) N 2 has a larger binding energy than N 2 + , but O 2 + has a larger binding energy than O 2 . Explain briefly. 9) The vibrational states of diatomic molecules are quantized, and transitions between different vibrational states can be induced by incident photons. Would the photon emitted by the H 2 molecule when falling from its first excited vibrational state to its vibrational ground state be of higher or lower energy than the corresponding photon for the H 2 + molecular ion? Why? 10) Compare the bond properties of the diatomic molecules, H 2 + , H 2 , Li 2 + and Li 2 , based on what you can infer from their molecular orbital total energy curve as a function of interatomic separation distance.
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