Chemistry 347 2014 Problem Set #1 Due: September 19, 2014 1. The gradient ⃗ of a function f is defined as: ⃗ ̂( ) ̂( ) where ̂ and ̂ are unit vectors pointing along the x and y axis, respectively. Find the gradient of the scalar function: ( x 1) 2 y 2 f log ( x 1) 2 y 2 2. In three-dimensional space, a classical particle moves in a potential field (in Joules, J) given by: V ( x, y, z) ax 3 by 3 cz 3 dxy exz fyz , where a, b, c, d, e, and f are constants. When the particle is at point x = 1 cm, y = 2 cm, and z = 3 cm, what are the x-, y- and zcomponents of force (in Newton, N) acting on it in terms of constants of the problem? 3. a) Calculate the de Broglie wavelengths (in μm = 10-6 m) of a 3000 lb car traveling 55 miles per hour and of an electron also traveling 55 miles per hour. b) Determine the wavelength of an electron accelerated by a 100 V potential difference (note: the electron energy in eV = the kinetic energy). c) What kinetic energy must a hydrogen atom have so it`s de Broglie wavelength is the size the atom in its 1s orbital (use size = 0.529 Å; one Angstrom, Å = 10-10 m). 4. Newton’s equation of motion for a particle of mass moving freely in the +ve x-direction is: Fx m d 2x 0 dt 2 a) Find the position as a function of time, x(t), subject to the following initial conditions: position x = xo and velocity v =vo at time t = 0. CHEM 347 2014 Problem Set #1-1 b) What is the momentum of the particle, px, as a function of time. (Note: px and x are well determined in classical mechanics, but as we shall see, are not in quantum mechanics). c) Now imagine an elastic collision which reverses the particle’s direction, and its velocity, +v to –v. Is the kinetic energy of the particle conserved? Why or why not? 5. Reduce the following complex numbers to its simplest form: a) (3+i)+(2-5i) b) (2-i)(2+i) c) (2+i)(3-2i) d) (1+i)/(1-i) e) (1+i)3 f) 1/i g) i15 + i9 + i h) √i) (x+iy)3 6. The needle on a dial is free to swing between two pins so that upon giving the needle a flick it could adopt an angle from the horizontal between θ = 0 and 2π/3. a) Recognizing that the probability could be zero sketch the probability density ρ(θ) from –π/2 to 3π/2 where ρ(θ)dθ is the probability that the needle will come to rest after a flick between θ and θ+dθ. Explain. b) Normalize ρ(θ) c) Use the result in b) to numerically calculate <θ>, <cos(θ)>, and <sin2(θ)>. CHEM 347 2014 Problem Set #1-2 7. Imagine you are at a carnival and observe a dart game in which you throw a single dart at the following target: The dollar amounts indicate how much money you will win if you hit the corresponding part of the target with your dart. The geometry is such that the radius r of each circle increases linearly between successive areas; that is, the inner circle has a radius of r, the middle circle, 2r, and the outer circle 3r. Assuming you will always hit the target: a) What is the probability you will hit the red area; the blue area, and the green area? b) How much on average will the carnival pay out if the prize money is as indicated in the picture? Give that each throw will cost you $1.50, are the odds in your favor or do they favor the carnival? CHEM 347 2014 Problem Set #1-3

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