Contemporary Physics I – HW 8HW 8Due November 30, 2007Note the 2 week due datePlease answer all questions clearly and concisely. While you need not transcribe the questioncompletely, it s hould be clear from your an s wer alone what you are talking about.You are strongly encouraged to discuss the homework with your classmates, but you must completethe written homework by yourself, and of cours e, the material you submit must be your own.Remember, show all of your work!1. Compute the DeBroglie wavelength of the following(a) An electron traveling at 20m/s(b) A 70 kg human with a kinetic energy of 8000 J.(c) A pr oton with γ = 2.5.2. You observe three (non-interacting) particles with:m1=1 kg, m2=2 kg, and m3=3 kg, and at positions:~r1=< 2, 1, 0 > m, ~r2=< − 4, 2, 0 > m, ~r3=< 1, 0, 0 > m, and with velocities:~v1=< 3, 0, 0 > m/s, ~v2=< − 2, 2, 0 > m/s, ~v3=< − 1, 3, 0 > m/s.(a) Wh ere is the center of mass? (I’m looking for a vector.)(b) Wh at is the center of mass velocity?(c) What is the translational kinetic energy?(d) Wh at is the total energy of the system (not including mass energy)?3. Consider a wave with a wavelength λ = 500nm, and a propagation speed 3 × 108m/s (youmay know this as c).(a) Wh at is the wavenumber of wave (k)?(b) Wh at is the frequency of oscillations (ν)?(c) If the wave is a lightwave, at what en ergy is it?(d) Wh at is the momentum of the corresponding photon?4. A quantum mech anical particle is in the following potential:U(x) =12mω2x2(You may recognize this as a simple way of re-writing the energy of a harmonic oscillator). Iassert to you that the Schroedinger Wave Equation:−¯h22md2ψdx2= (E − U(x))ψcan be solved in the ground state as:ψ(x) = ψ0exp(−x2/2σ2)(a) Using maple (if you like – in wh ich case, please include a printout), or with pen andpaper, show that my solution for the wavefunction is correct.(b) P lease plot the solution.(c) What is the value of σ that has to be put into the s olution?(d) Wh at is the energy of th is