Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton for two nonnteractng dentcal partcles n a box, (9-1). 9-0 By applyng the technque of separaton of varables, show that, for a potental of the addtve form of (9-), there are solutons to the two-partcle tme-ndependent Schroednger equaton, (9-1), n the product form of (9-3). 9-03 Exchange the partcle labels n the two probablty densty functons, obtaned from the symmetrc and antsymmetrc egenfunctons of (9-8) and (9-9), and show that nether s affected by the exchange. The probablty denstes are 9-04 Verfy that expanded from of the three-partcle egenfuncton of Example 9- s antsymmetrc wth respect to an exchange of the labels of two partcles. 9-05 Verfy that expanded from of the three-partcle egenfuncton of Example 9- s dentcally equal to zero f two partcle are n the same space and spn quantum state. 9-06 Verfy that the 1 3! normalzaton factor quoted n example 9- s correct. 9-07 Verfy that the expanded from of the three-partcle egenfuncton of Example 9-3 s symmetrc wth respect to an exchange of the labels of two partcles. 9-08 An α partcle contans two protons and two neutrons. Show that f each of ts 1 / 6
consttuents s antsymmetrc then t must be symmetrc, as stated n Table 9-1. (Hnt : Consder a par of α partcles, and the effect of exchangng the labels of all the consttuents n one wth those of all the consttuents n the other.) 9-09 Wrte an expresson for the expectaton value of the energy assocated wth the Coulomb nteracton between the two electrons of a helum atom n ts ground state. Use a space egenfuncton for the system composed of products of one-electron atom egenfunctons, each of whch descrbes an electron movng ndependently about the Z = nucleus. Do not bother to evaluate the expectaton value ntegral, but nstead comment on ts relaton to the energy levels shown n Fgure 9-7. 9-10 Prove that any two dfferent nondegenerate bound egenfunctons ψ ( x) and ψ ( x) that are solutons to the tme-ndependent Schroednger equaton for the same potental V( x ) obey the orthogonalty relaton ψ * ( x) ψ( xdx ) = 0 (Hnt : () Wrte the equatons to whch ψ ( x) and ψ ( x) are solutons, and then take the complex conugate of the second one to obtan the equaton satsfed by the equaton n * ψ. () Multply the equaton n ψ by ψ *, * ψ by ψ, and then subtract. () Integrate, usng a relaton such * * * d ψ d ψ d * dψ dψ as ψ ψ ( ) = ψ ψ.) The proof can be extended to dx dx dx dx dx nclude degenerate egenfunctons, and also unbound egen functons that are properly normalzed. Can you see how to do ths? 9-11 (a) By gong through the procedure ndcated n Secton 9-5, develop the tme-ndependent Schroednger equaton for a system of z electrons of an atom movng ndependently n a set of dentcal net potentals V() r. (b) Then separate t nto a set of Z dentcal tme-ndenendent Schroednger equatons, one for each electron. (c) Verfy that the form of a typcal one s as stated n (9-). (d) / 6
Compare ths form wth the tme-ndependent Schroednger equaton for a one-electron atom, (7-1). 9-1 (a) Show that there are N! terms n the lnear combnaton for an antsymmetrc total egenfuncton descrbng a system of N ndependent electrons. (Hnt : Consder Example 9-, and use the mathematcal technque of nducton.) (b) Evaluate the number of such terms for the case of the argon atom wth Z = 18. (Hnt : Use a mathematcal table to evaluate N!, or use Strlng s formula, found n most mathematcal references, to approxmate t.) (c) State brefly the connecton between the results of (b) and the procedure used by Hartree to treat the argon atom. 9-13 (a) Use nformaton from Fgure 9-11 to make a sketch, on semlog paper, of the net potental V() r for the argon atom. Be sure to determne several values for r between 0 and 0.5, as ths nformaton wll be used n Problem 18. (b) Also a0 show the energy levels E 1 and E, usng estmates from Example 9-5, and energy level E 3, usng measured data from Fgure 9-15. 9-14 (a) Fnd the value of Z 1 for the helum atom whch, when used n the energy equaton, (9-7) leads to agreement wth the ground state energy shown n Fgure 9-6. (b) Compare Z 1 wth Z. (c) Is Z 1 meanngful for an atom wth as few electrons as helum? Explan brefly..4 9-15 From Fgure 9-6 estmate the average dstance between the two electrons n a helum atom (a) n the ground state and (b) n the frst excted state. Neglect the exchangeenergy. (a) From Fg. 9-6, E =+ 30eV E coul 1 e = = 4πε r 0 9 30 = (9 10 ) r = 0.048nm coul (1.6 10 ) 19 19 r(1.6 10 ) 3 / 6
(b) E =+ 9eV r = 0.16nm ## coul 9-16 (a) Use the Z n for the argon atom obtaned n Example 9-5 n the one-electron atom equaton for the radal coordnate expectaton value, to estmate the rad of the n = 1,, and 3 shells of the atom. (b) Compare the results wth Fgure 9-10. 9-17 Develop a mathematcal argument for the tendency, llustrated n Fgure 9-1, of an atomc electron wth angular momentum L to avod the pont about whch t rotates. Treat the electron semclasscally by assumng that t moves around an orbt n a fxed plane passng through the nucleus. (a) Show that ts total energy p// L p// can be wrtten E = + [ V( r) + ] = + V ( r) where p // s ts m mr m component of lnear momentum parallel to ts radal coordnate vector of length r. (b) Explan why ths ndcates that ts radal moton s as t would be n a one-dmensonal system wth potental V () r. (c) Then show that V () r become L repulsve for small r because of the domnant behavor of the term mr, sometmes called the centrfugal potental. 9-18 (a) Sketch the potentals V () r for the argon atom wth l = 0 and l = 1, defned n Problem 17, by addng the correspondng centrfugal potentals to the V() r obtaned n Problem 13. (b) Also sketch the energy level E. (b) Show the classcal lmts of moton, wthn whch E V () r. (d) Compare these lmts wth the radal probablty denstes of Fgure 9-10, for n =, l = 0, and n =, l = 1. 9-19 Wrte the confguratons for the ground states of 8 N, 9 Cu, 30 Zn, 31 Ga. 9-0 Wrte the confguratons for the ground states of all the lanthandes, makng as much use as possble of dtto marks. 9-1 Recent work n nuclear physcs has led to the predcton that nucle of atomc 4 / 6
number Z = 110 mght be suffcently stable to allow some of the element Z = 110 to have survved from the tme the elements were created. (a) Predct a lkely confguraton for ths element. (b) Make a predcton of the chemcal propertes of the element. (c) Where would be a lkely place to start searchng for traces of t? 9- (a) From nformaton contaned n Fgure 9-6 and 9-15, determne the energy requred to remove the remanng electron from the ground state of a sngly onzed helum atom. (b) Compare ths energy predcted by the quantum mechancs of one-electron atoms. 9-3 (a) Draw a schematc representaton of a standard energy-level dagram for the T atom, showng the states populated by electrons for a case n whch one electron s mssng from the K shell. The dagram should be comparable to the one n Fgure 9-9 n that t should not attempt to gve the energes of the levels to an accurate scale, and no dstncton should be made between L I, L II, and L III levels, etc. (b) Do the same for a case n whch one electron s mssng from the L shell. (c) Draw a schematc representaton of an x-ray energy-level dagram showng the energes of the atom when a hole s n the K or L shell. (d) Compare the utlty of the standard and x-ray energy-level dagrams for cases n whch a hole s n an nner shell. (e) Also make such a comparson for cases n whch a hole s n an outer shell. 9-4 The wavelengths of the lnes of the K seres of 74 W are (gnorng fne structure) : for K α, λ = 0.10 Å; for K β, λ = 0.184 Å; for K γ, λ = 0.179 Å. The wavelength correspondng to the K absorpton edge s λ = 0.178 Å. Use ths nformaton to construct an x-ray energy-level dagram for 74 W. 9-5 (a) Make a rough estmate of the mnmum acceleratng voltage requred for an x-ray tube wth a 6 Fe anode to emt a L α lne of ts spectrum. (Hnt : As n Example 9-5, Z Z 10.) (b) Also estmate the wavelength of the L α photon. 870V 5 / 6
9-6 (a) Use Moseley s data of Fgure 9-18 to determne the values of the constants C and a n ts emprcal formula, (9-31). (b) Compare these values wth those of (9-30), whch was derved from the results of the Hartree theory. 6 1 (a) 8.65 10 m, 1.7 9-7 It s suspected that the cobalt s very poorly mxed wth the ron n a block of alloy. To see regons of hgh cobalt concentraton, an x-ray s taken of the block. (a) Predct the energes of the K absorpton edges of ts consttuents. (b) Then determne an x-ray photon energy that would gve good contrast. That s, determne an energy of the photon for whch the probablty of absorpton by a cobalt atom would be very dfferent from the probablty of absorpton by an ron atom. (a) Co :8.50keV, Fe:7.83keV (b) 8.50keV 8 9-8 The Lyman-alpha lfetme n hydrogen s about 10 sec. From ths, fnd the lfetme for the K α x-ray transton n lead. (Hnt : For the nner electron n lead 1 the wavefunctons are hydrogenc wth approprate effectve Z; lfetme =, see R (8-43).) 16.44 10 sec 6 / 6