Name _________________ Test 2 October 17, 2012 This test consists of three parts. Please note that in parts II and III, you can skip one question of those offered. The equations below may be helpful with some problems. Constants h 6.626 10 J s 4.136 1015 eV s Black Bodies 34 1.055 1034 J s 6.582 1016 eV s k B 1.3807 1023 J/K 8.6173 105 eV/K k 8.988 10 N m / C e 1.602 1019 C 1 eV 1.602 1019 J ke 2 1 7.29735 103 137 c Compton Effect Wave h Relationships 1 cos mc 2 h 12 k 2.426 10 m mc 1 f 2 T Hydrogen Spectrum 9 1 1 2 2 n m 91.17 nm 2 2 1 U 2 k BT 15 c 4 3 maxT 2.898 103 m K Rutherford Scattering kqQ cot b 2 m v 2 R 2Ze 2 k E Hydrogen-Like Atoms c2 2 Z 2 k 2e4 Z 2 E 2 2 n 2 2n 2 13.6 eV Z 2 E n2 Reduced Mass mM mM Part I: Multiple Choice [20 points] For each question, choose the best answer (2 points each) 1. If the wave function is given by x 2ie x , what is the probability density? A) 4e 2 x B) 4ie 2 x C) 4e 2 x 2. The formula for group velocity is A) f B) k C) k D) 4e x 2 D) d dk E) 4e x 2 E) f 3. In the Frank-Hertz experiment, what is the source of the light that is being emitted? A) The voltage is getting large enough to cause arcing, which is causing the glow B) The accelerated electrons are getting converted directly into photons C) Accelerated electron pairs collide to produce light D) The electrons collide with the anode, which becomes warm and glows E) The atoms that have been impacted by electrons fall back to a lower energy state 4. Why do different isotopes of hydrogen have actually very slightly different spectral lines? A) The charges of the nuclei are very slightly different B) You have to work with the reduced mass , rather than the electron mass, which is a tiny bit different for different isotopes C) The photons emitted are partially absorbed by the nucleus, losing some energy D) The atoms are in motion, causing Doppler shift, but the heavier isotopes are moving more slowly E) Magnetic interactions with the nucleus cause a shift in the energies 5. Suppose you know the speed of an electron. What can you learn by studying the curve it makes in a magnetic field? A) Its mass m B) Its charge e C) The ratio of its mass and its charge, e/m D) The product of its mass and its charge, me E) None of these can be learned from this experiment 6. The fundamental assumption Planck had to make to explain the black body spectrum was A) Light acts as if it comes in chunks of energy proportional to the frequency B) Light acts as if it comes in chunks of energy proportional to the wavelength C) The black body spectrum must have a peak wavelength max D) The angular momentum of photons was always a multiple of E) Light is actually waves, not particles as previously thought 7. Rutherford scattering with -particles allowed Rutherford to A) Measure the angular momentum of the electrons in an atom B) Measure the energy levels of the electrons in an atom C) Measure the charge of the electrons in an atom D) Measure the size of atoms E) Measure the size of the nucleus 8. Which of the following was a success of the Bohr model of the atom? A) It correctly predicted the approximate radius of hydrogen (only) B) It correctly predicted the wavelengths of the light emitted from hydrogen (only) C) It correctly predicted the wavelength of light emitted from other atoms, like neutral helium (only) D) A and B are true, but not C E) A, B, and C are all true 9. What is the correct relationship between h and ? h 1 2 A) 2 h B) C) D) h 2 h E) 1 2 h 10. How did deBroglie’s relation p h help explain the Bohr model? A) It explained diffraction for electrons, which was assumed by Bohr B) It explained why only circular orbits worked in the Bohr model C) It explained why energy always was emitted as a single photon D) It explained why the angular momentum was always multiples of E) It explained the isotope effects of different isotopes of hydrogen Part II: Short answer [20 points] Choose two of the following questions and give a short answer (2-3 sentences) (10 points each). 11. Explain qualitatively how one can use the spectrum of light from a distant star to determine the star’s temperature, assuming that spectrum is a black body distribution. 12. When a photon scatters from an electron at rest, how does the scattered photon’s wavelength, momentum, and energy differ (increase or decrease) from the initial photon? 13. Classically, it was not understood why the electron did not simply fall to the position of the nucleus in an atom. Explain, in terms of the uncertainty principle, why this doesn’t happen. Part III: Calculation: [60 points] Choose three of the following four questions and perform the indicated calculations (20 points each). 14. Ytterbium (Yb) has a work function 2.60 eV . (a) What is the lowest frequency f that can liberate an electron from ytterbium? (b) Suppose light with wavelength 257 nm impacts some ytterbium. What is the voltage Vmax that the liberated electrons from ytterbium can overcome? (c) When an unknown light is shone on ytterbium, it is found that the electrons can only overcome a voltage of Vmax = 1.75 V. What is the frequency of the light? 15. A certain element, when all but one electron has been removed, emits light with a wavelength of 7.60 nm when an electron falls from n = 2 to n = 1. (a) What is the energy of this photon? (b) What is the Z-value of this element? (c) If this atom had an electron fall from n = 9 to n = 8, what would be the energy of the resulting photon? 16. Neutrons with a mass of m 1.675 1027 kg are being fired at a neutrons narrow slit of width w 14.0 nm with a kinetic energy of E 4.86 1023 J . (a) What is the momentum of these neutrons? (b) By Carlson’s rule, what is the uncertainty in their vertical position based on the fact that they went through the slit? Based on the uncertainty principle, what is the corresponding uncertainty in their vertical momentum p ? (c) Assume the total momentum p is the same after they pass through the slit, but they have now acquired a vertical momentum p . How much has their direction changed by passing through the slit? 17. The wave function of a particle is given by N a 3 x x 4 0 x a, x 0 otherwise . Na 4 where N and a are positive constants. The wave is sketched at right. x a (a) What is the most likely place to find the particle? (b) What is the normalization constant N? (c) If the particle’s position is measured, what is the probability that it will be found to have a value in the range 0 x a 3 2 ?