November 30, 2016 Notre Dame University (FNAS) Course PHS 213 (Modern Physics) Academic year 2016-2017 Fall semester Instructor: Dr. Youssef K. El-Hage Quiz #2 (Duration: 1 h) 1) Consider a singly ionized Helium atom (Z=2) according to the Bohr model. a) What are the three lowest allowed energies (E1, E2, and Ez) of the electrons (in eV)? b) What are the energies of the emitted photons when the electron undergoes transitions from the third level Ez to each of the lower energy levels? c) Which of these transitions results in the highest frequency photon? What is this frequency? (20 pts) 2) An electron, not localized in space, is described by a wavelike solution of the Schrödinger equation. Suppose that a free electron (no potential acting on it), with a non-relativistic kinetic energy of 1000 eV, has a wave-function given by '(x) = A sin(kx – wt), where k is the wavenumber = 27/2) and the angular frequency (= 271f). a) What is the relation between the wave number k and the kinetic energy K? (Hint: Use the De Broglie's wavelength formula and the formula for K in terms of the momentum p) b) Find the values of k and w for this electron wave (Hint: Kinetic energy K = hf) c) What is the value of the phase velocity of this electron wave? d) If the kinetic energy is that of a classical particle, what would be the value of this velocity? Is it the same as the one calculated in c)? (20 pts) 3) A particle of mass m is contained in a one-dimensional infinite well (a box) of width L. Find the probability that the particle will be located in the region L/3 < x < 2L/3 for the fourth a) energy level. e 2.6 eV 1 2 b) Sketch the distributions of the wave function Y and of its corresponding probability density for the cases n= 1, 2 and 3. (30 pts) 4) A free electron of kinetic energy 2.5 eV approaches a region of constant potential energy 2.6 eV (region 2). a) What are the wave number k, and wavelength ay of the electron in the free region (region 1)? b) In region 2, does the electron propagate harmonically or does its wave function decrease exponentially? If it propagates harmonically, what is the wave number k2 of the electron in this region? If it decays away exponentially, what is the decay constant a in the expression e-c? (20 pts) 5) Find the wavelength of the spectral line that corresponds to a transition in hydrogen from the n=10 state to the ground state. To what part of the electromagnetic spectrum does it correspond? (10 pts) x even GOOD LUCK Open-book exam (only official textbooks are allowed)
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