Quantum Physics

Problems 935tum number n. (e) Should you expect to observequantum effects in the Earth–Sun system?22. An electron is in the nth Bohr orbit of the hydrogenatom. (a) Show that the period of the electron isT t o n 3 , and determine the numerical value of t o .(b) On the average, an electron remains in the n 2orbit for about 10 s before it jumps down to then 1 (ground-state) orbit. How many revolutionsdoes the electron make before it jumps to theground state? (c) If one revolution of the electron isdefined as an “electron year” (analogous to anEarth year being one revolution of the Eartharound the Sun), does the electron in the n 2 orbit“live” very long? Explain. (d) How does theabove calculation support the “electron cloud” concept?23. Consider a hydrogen atom. (a) Calculate the frequencyf of the n 2 : n 1 transition, and compareit with the frequency f orb of the electron orbitalmotion in the n 2 state. (b) Make the samecalculation for the n 10 000 : n 9 999 transition.Comment on the results.24. Two hydrogen atoms collide head-on and endup with zero kinetic energy. Each then emits a121.6-nm photon (n 2 to n 1 transition). Atwhat speed were the atoms moving before thecollision?25. Two hydrogen atoms, both initially in the groundstate, undergo a head-on collision. If both atoms areto be excited to the n 2 level in this collision,what is the minimum speed each atom can have beforethe collision?26. (a) Calculate the angular momentum of the Moondue to its orbital motion about the Earth. In yourcalculation, use 3.84 10 8 m as the averageEarth–Moon distance and 2.36 10 6 s as the periodof the Moon in its orbit. (b) If the angularmomentum of the moon obeys Bohr’s quantizationrule ( L n), determine the value of the quantumnumber n. (c) By what fraction would theEarth–Moon radius have to be increased to increasethe quantum number by 1?Section 28.4 Modification of the Bohr TheorySection 28.5 De Broglie Waves andthe Hydrogen Atom27. (a) Find the energy of the electron in the groundstate of doubly ionized lithium, which has anatomic number Z 3. (b) Find the radius of itsground-state orbit.28. (a) Construct an energy level diagram for the He ion, for which Z 2. (b) What is the ionization energyfor He ?29. The orbital radii of a hydrogen-like atom is given bythe equationWhat is the radius of the first Bohr orbit in (a) He ,(b) Li 2 , and (c) Be 3 ?30. (a) Substitute numerical values into Equation 28.19to find a value for the Rydberg constant for singlyionized helium, He . (b) Use the result of part (a)to find the wavelength associated with a transitionfrom the n 2 state to the n 1 state of He . (c)Identify the region of the electromagnetic spectrumassociated with this transition.31.r n2 2Zm e k e e 2Determine the wavelength of anelectron in the third excited orbit of the hydrogenatom, with n 4.32. Using the concept of standing waves, de Brogliewas able to derive Bohr’s stationary orbit postulate.He assumed that a confined electron couldexist only in states where its de Broglie waves formstanding-wave patterns, as in Figure 28.10a. Considera particle confined in a box of length L to beequivalent to a string of length L and fixed at bothends. Apply de Broglie’s concept to show that(a) the linear momentum of this particle is quantizedwith p mv nh/2L and (b) the allowedstates correspond to particle energies of E n n 2 E 0 ,where E 0 h 2 /(8mL 2 ).Section 28.6 Quantum Mechanics andthe Hydrogen AtomSection 28.7 The Spin Magnetic QuantumNumber33. List the possible sets of quantum numbers for electronsin the 3p subshell.34. When the principal quantum number is n 4,how many different values of (a) and (b) m arepossible?35. The -meson has a charge of e, a spin quantumnumber of 1, and a mass 1 507 times that of theelectron. If the electrons in atoms were replaced by -mesons, list the possible sets of quantum numbersfor -mesons in the 3d subshell..