 # Problem 1: If electromagnetic radiation with a photon energy of 20 ...

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Problem 1: If electromagnetic radiation with a photon energy of 20 meV impinges on two slits spaced 0.1 mm apart, find the angle 0 between the center line and the 1st minimum and 1st maximum of the resulting interference pattern. Problem 2: A non-relativistic beam of electrons travels at 5% the speed of light and impinges on a slit that is 0.05 um wide. Use modern units (e.g. eV, etc) to find the angle 0 between the center line and 1st minimum of the resulting diffraction pattern. How far away would you locate a detector so that the 1st minimum is spaced 2 mm from the center line? Problem 3: If a Cu Ka x-ray diffracts from a periodic crystal and the first maximum is detected by a Geiger counter at 20 = 40°, then find the crystal spacing d of the crystal. Problem 4: The wave function for a free electron is given by y (x) = Acos(3x-t) where the units of X are in nanometers. Find the wavenumber k (nm-¹), de Broglie wavelength A, momentum term pc (in eV), and kinetic energy Ek (eV) of the electron (use "modern" units). Problem 5: Use the de Broglie relation to find A and k for electrons with kinetic energy of 300eV. Problem 6: If an electron moves between two barriers located 0.1 nm apart, find its minimum energy (eV) due to the Heisenberg Uncertainty principle. To increase this energy by a factor of 9, what should the new barrier distance be? Problem 7: When a beam of electrons travels through a slit, there is an uncertainty in the lateral position of the beam (i.e.slit width 4y) and an accompanying minimum uncertainty in the transverse momentum Apy due to Uncertainty Principle. This momentum uncertainty results in a spreading of the beam AY, where the spread depends on the distance from the slit. Find the spread width of a beam traveling at Vx = 106 m/s through a 100-m wide slit when it hits a plate located D = 50 m past the slit. Problem 8: A proton is known to be within an interval 6 fm (the radius of a large nucleus). Roughly what is the minimum uncertainty in its velocity? (Treat this problem as 1 -dimensional, and express your answer as a fraction of c. Problem 9: An excited state of a certain nucleus has a lifetime of 5 X 10-18 S. Find the minimum possible uncertainty in its energy. Problem 10: Use Excel (or similar program) to graph the energy VS. momentum (E VS. pc) relationship for a photon and a He atom (3728 MeV/c2). Include the formula for E VS. pc on each graph and use pc values from 0 to 1000 eV. Next, derive the phase and group velocities of each particle. Which particle shows no dispersion?

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