Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations

Transcription

1 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) 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 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 Verfy that the 1 3! normalzaton factor quoted n example 9- s correct 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 An α partcle contans two protons and two neutrons. Show that f each of ts 1 / 6

2 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 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

3 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 (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 (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 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 = (9 10 ) r = 0.048nm coul ( ) r( ) 3 / 6

4 (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 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 (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 = 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

5 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 = Å; for K γ, λ = Å. The wavelength correspondng to the K absorpton edge s λ = 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

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) m, 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 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).) sec 6 / 6