Foundation of Modern Physics Homework #3 Due by: 2015.05.19. (Tue) 12:30 #1. The Stanford Linear Accelerator accelerated electrons to an energy of 50 GeV. What is the de Broglie wavelength of these electrons? What fraction of a proton’s diameter (d ≈ 2 × 10-15m) can such a particle probe? #2. Davisson and Germer performed their experiment with a nickel target for several electron bombarding energies. At what angles would they find diffraction maxima for 48-eV and 64-eV electrons? #3. Consider electrons of kinetic energy 6.0 eV and 600 keV. For each electron, find the de Broglie wavelength, particle speed, phase velocity (speed), and group velocity (speed). #4. Two waves are traveling simultaneously down a long Slinky. They can be represented by Ψ1 (x, t) = 0.0030 sin(6.0x – 300t) and Ψ2 (x, t) = 0.0030 sin(7.0x-250t). Distances are measured in meters and time in seconds. (a) Write the expression for the resulting wave. (b) What are the phase and group velocities? (c) What is ∆x between two adjacent zeros of Ψ? (d) What is ∆k∆x? #5. Show that the uncertainty principle can be expressed in the form ∆L∆θ ≥ /2, where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined? #6. A wave function Ψ is A(eix + e-ix) in the region –π<x<π and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x = 0 and x = π/8, and (b) between x = 0 and x = π/4. #7. For the infinite square-well potential, find the probability that a particle in its ground state is in each third of the one-dimensional box: 0≤x≤L/3, L/3≤x≤2L/3, 2L/3≤x≤L. Check to see that the sum of the probabilities is one. #8. Find the energies of the second, third, fourth, and fifth levels for the three-dimensional cubical box. Which energy levels are degenerate? #9. Show that the energy of a simple harmonic oscillator in the n = 1 state is 3ℏω/2 by substituting the wave function ψ1 = / directly into the Schrödinger equation. #10. A 1.0-eV electron has a 2.0 × 10-4 probability of tunneling through a 2.5-eV potential barrier. What is the probability of a 1.0-eV proton tunneling through the same barrier?