Planck Radiation Law - Introduction to Quantum Mechanics - Exam, Exams of Quantum Mechanics

Download Planck Radiation Law - Introduction to Quantum Mechanics - Exam and more Exams Quantum Mechanics in PDF only on Docsity! PhD Candidacy Exam Fall 2004 – Quantum Mechanics Choose any 3 of 4. Problem 1. (a) Planck’s radiation law is given by uω = ω2 c3π2 h̄ω exp ( h̄ω kBT ) − 1 . (1) Show that the energy density uω in terms of wavelength becomes uλ = 8πhc λ5 1 exp ( hc λkBT ) − 1 . (2) (b) Find the wavelength for which the energy distribution is maximum (assume that hc/λkBT is large enough, so that e−hc/λkBT → 0). The relation Tλ =constant is known as Wien’s Law. (c) Derive the Stefan-Boltzmann law from Planck’s law, using ∫ ∞ 0 x3 ex − 1dx = π4 15 = 6.4938. (3) Calculate the value of the Stefan-Boltzmann constant. Problem 2. (a) Let us consider the harmonic oscillator whose Hamiltonian is given by H = (a†a + 1 2 )h̄ω. (4) By using a†a|n〉 = n|n〉 (5) and the commutation relations of the operators [a, a†] = 1 (6) show that the wavefunction a†|n〉 is proportional to the wavefunction |n + 1〉. (b) The wavefunctions for the harmonic oscillator are given by |n〉 = 1√ n! (a†)n|0〉. (7) Determine the constant for which a†|n〉 = constant|n + 1〉. (c) We can also write x̂ and p̂x in terms of the creation and annihilation operators x̂ = √ h̄ 2mω (a† + a) and p̂x = i √ mh̄ω 2 (a† − a). (8) By determining the values of 〈x̂2〉 and 〈p̂2x〉, show that Heisenberg’s uncertainty principle is satisfied. Problem 3. (a) An electron is harmonically bound at a site. It oscillates in the x direction. The solutions of this harmonic oscillator are H = h̄ω(a†a + 1 2 ). (9) We introduce a perturbation by an electric field created by a positive point charge at a distance Rx̂. Show that the disturbing potential can be written as (you can leave out the constant energy shift from the electric field) H ′ = P (a + a†) with P = e2 4π²0R2 √ h̄ 2mω (10) when R is much larger than the amplitude of the oscillation. (b) Note that the Hamiltonian H + H ′ is not diagonal, since different values of n are coupled with each other. Show that the total Hamiltonian can be diagonalized by adding a constant shift to the step operators. (c) What is the shift in energy as a result of the perturbation H ′ ? Problem 4. Two similar metals are separated by a very thin insulating layer along the plane x = 0. The potential energy is constant inside each metal; however, a battery can be used to establish a potential difference V1 between the two. Assume that the electrons have a strong attraction to the material of the insulating layer which can be modeled as an attractive delta function at x = 0 for all values of y and z. A sketch of the potential energy along the x direction is shown in Figure 1. Here S and V1 are positive. (a) Assume that the metals extend to infinity in the y and z directions. Write down the correct three-dimensional functional form for an energy eigenfunction of a state bound in the x direction. Sketch its x dependence. (b) Find the maximum value of V1 for which a bound state can exist. Express your answer in terms of h, m, and S. (c) Find the energy of the bound state in terms of h, m, S, and V1. Show that your answer is consistent with 4(b).