Quantum Physics

Problems 93748. A dimensionless number that often appears inatomic physics is the fine-structure constant, where k e is the Coulomb constant.(a) Obtain a numerical value for 1/. (b) In termsof , what is the ratio of the Bohr radius a 0 to theCompton wavelength C h/m e c ? (d) In terms of, what is the ratio of the reciprocal of the Rydbergconstant 1/R H to the Bohr radius?49. Mercury’s ionization energy is 10.39 eV. The threelongest wavelengths of the absorption spectrum ofmercury are 253.7 nm, 185.0 nm, and 158.5 nm.(a) Construct an energy-level diagram for mercury.(b) Indicate all emission lines that can occur whenan electron is raised to the third level above theground state. (c) Disregarding recoil of the mercuryatom, determine the minimum speed an electronmust have in order to make an inelastic collisionwith a mercury atom in its ground state.50. Suppose the ionization energy of an atom is 4.100eV. In this same atom, we observe emission linesthat have wavelengths of 310.0 nm, 400.0 nm, and1 378 nm. Use this information to construct the energy-leveldiagram with the least number of levels.Assume the higher energy levels are closer together.51.A laser used in eye surgery emits a3.00-mJ pulse in 1.00 ns, focused to a spot 30.0 min diameter on the retina. (a) Find (in SI units) thepower per unit area at the retina. (This quantity iscalled the irradiance.) (b) What energy is deliveredper pulse to an area of molecular size—say, a circulararea 0.600 nm in diameter.52. An electron has a de Broglie wavelength equal tothe diameter of a hydrogen atom in its ground state.(a) What is the kinetic energy of the electron?(b) How does this energy compare with the groundstateenergy of the hydrogen atom?53. Use Bohr’s model of the hydrogen atom to showthat, when the atom makes a transition from thestate n to the state n 1, the frequency of the emittedlight is given by k e e 2 /cf 2 2 mk e 2 e 4h 3 2n 1(n 1) 2 n 254. Calculate the classical frequency for the light emittedby an atom. To do so, note that the frequency ofrevolution is v/2r, where r is the Bohr radius.Show that as n approaches infinity in the equationof the preceding problem, the expression giventhere varies as 1/n 3 and reduces to the classical frequency.(This is an example of the correspondenceprinciple, which requires that the classical andquantum models agree for large values of n.)55. A pi meson ( ) of charge e and mass 273 timesgreater than that of the electron is captured by ahelium nucleus (Z 2) as shown in FigureP28.55. (a) Draw an energy-level diagram (in unitsof eV) for this “Bohr-type” atom up to the first sixenergy levels. (b) When the -meson makes atransition between two orbits, a photon is emittedthat Compton scatters off a free electron initiallyat rest, producing a scattered photon of wavelength 0.089 929 3 nm at an angle of 42.68°, as shown on the right-hand side of FigureP28.55. Between which two orbits did the -mesonmake a transition?n in f“Pi mesonic” He + atom(Z = 2, m p = 273m e )lFigure P28.55Free electron56. When a muon with charge e is captured by a proton,the resulting bound system forms a “muonicatom,” which is the same as hydrogen, except witha muon (of mass 207 times the mass of an electron)replacing the electron. For this “muonicatom,” determine (a) the Bohr radius and (b) thethree lowest energy levels.57. In this problem, you will estimate the classical lifetimeof the hydrogen atom. An accelerating chargeloses electromagnetic energy at a rate given by 2k e q 2 a 2 /(3c 3 ), where k e is the Coulomb constant,q is the charge of the particle, a is its acceleration,and c is the speed of light in a vacuum. Assume thatthe electron is one Bohr radius (0.052 9 nm) fromthe center of the hydrogen atom. (a) Determine itsacceleration. (b) Show that has units of energyper unit time and determine the rate of energy loss.(c) Calculate the kinetic energy of the electron anddetermine how long it will take for all of this energyto be converted into electromagnetic waves, assumingthat the rate calculated in part (b) remains constantthroughout the electron’s motion.58. An electron in a hydrogen atom jumps from someinitial Bohr orbit n i to some final Bohr orbit n f , as inOul'