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MIT 3 23 - Quantum Mechanics - exercice sheet 1

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MIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Quantum Mechanics - exercice sheet 1 September 12, 2007 1 In a monoatomic gas, one measure of the ”average speed” of a gas particle is � 2 �1/2 � 3kB Tthe root mean square speed defined as follow: vrms = v = m , where kB is the Boltzmann constant, T the temperature, and m the mass of a particle. Using this formula, calculate the De Broglie wavelength for Helium (He) and Argon (Ar) atoms at 100K and 500K. *** Datas: Helium molar mass, 4.033 g/mole Argon molar mass, 39.95 g/mole *** 2 Electrons have been used to determine molecular and solid structures using diffraction. Calculate the speed of an electron for which the De Broglie wave- length is equal to a typical bond length , namely, 0.150 nm. *** Datas: electron mass, 9.109 ∗ 10−31 kg *** 3 ¯hWhy can we conclude that the wave function ψ(x, t) = φ(x)e− iE t represents a standing wave? 4 If ψ(x, t) = A sin(kx − ωt) describes a wave travelling in the +x direction, how would you describe a wave travelling in the -x direction? 1 Nicolas Poilvert7 5 Distinguish between the following terms applied to the following set of func-tions, ψ1(x), ψ2(x), ..., ψn(x) : orthogonal, normalized and complete. Give a mathematical expression to express those terms using integrals. 6 Determine in each of the following cases if the function in the first column of table 1 is an eigenfunction of the opertor in the second column. If so, what is the corresponding eigenvalue? wavefunctions operators sin(φ) cos(φ) ∂ ∂φ e−x 2/3 ( 1 x ) d dx xy x ∂ ∂x + y ∂ ∂y 3 cos(θ)2 − 1 1 sin(θ) d dθ (sin(θ) d dθ ) x2 d dx Table 1: table of wavefunctions and operators dWhich of the following wavefunctions are eigenfunctions of the operator dx ? If they are eigenfunctions, what is the associated eigenvalue? ae−3x + be−3ix • sin2(x)• e−ix • • cos(ax) e−ix2 •


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