QUANTUM MECHANICS 2
QUANTUM MECHANICS 2 PHY 6646
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Date Created: 09/18/15
PHY 6646 K lngersent Degenerate RayleighSchrodinger Perturbation Theory Assume that we know the stationary states ofthe unperturbed Hamiltonian H07 namely the kets 777 satisfying Holmr enlmr The integer index 7 1 S r S 9 is used to distinguish among the gn eigenstates of energy 8 For simplicity7 we assume that the vector space is has a nite or countably in nite dimension The extension to continuous vector spaces is straightforward We seek stationary solutions My of the perturbed problem H0 l AH1lwnrgt Enrlwnrgt in the form of power series expansions um Emmi Dim lt2 j0 73970 Let us insert Eqs 2 into Eq 17 and collect terms having the same power of A At order A0 we have H0 7 07 which is satis ed by any linear combination of the unperturbed eigenkets of energy an ie7 9n W2gt7Zltcmgttln7tgt7 with E5227 t1 Orthonormality requires that zbfrlw 51Cn7cnst 6m At order A1 we nd H0 7 7 Acting from the left with m7 sl we obtain 9 8m 7 8n ltm7 6mnE77 CTLTS 7 Zltm75lH1ln7tgtCnrt t1 For m n7 Eq 3 yields 9 Zltn75lH1lnvtgtCnrt E l30nrs7 t1 which is the eigenequation for H1 in the gn dimensional subspace spanned by the unperturbed states of energy 8 It is perfectly consistent with the A0 result to choose the lwlggys to be the eigenkets of this problem We will assume henceforth that this is the case7 so that Mme we 7 an mm 4 where the rst order correction to the unperturbed energy is E53 7 WWW lt5 1 Note that we cannot assume that is an eigenket of H1 in the full vector space Equation 4 implies only that 9m mg Elam Z Ziw 31gtltw523tiH1iw zgt lt6 mnt1 For m 31 71 Eq 3 yields 9 m s H 0 ltm will ltmgjlf tgt cm lt En or lt0 lt0 ltw 2LW Zgt ltf i jwgt m e n lt7 lf 9 1 we can drop the second label for each eigenket Then Eqs 5 and 7 reduce to the standard results of nondegenerate perturbation theory Conversely it appears from Eqs 5 and 7 that the perturbative solution of the de generate problem to order A1 can be obtained from that of a nondegenerate problem by substituting a and Em a Em 251 However this conclusion is premature because Eq 3 does not determine nsl7 or alternatively We will now correct this omission Following a convention from the nondegenerate theory we enforce 3l nrgt 1 Thus 0 for all j gt 0 which includes as a special case ltw 2lw 2gt 0 To determine for s 31 7 it is necessary to proceed to order A2 in the expansion of Eq 1 Ho 7 EQWW E22 HDWED EEZWD Acting from the left with MEN we eliminate all but the term involving Then using the adjoint of Eq 6 with 5 replacing r we obtain 9m 0 E53 7 E22gtltMWZgt e Z Zw ilHllwiS ltw 2 w f x mnt1 where the last inner product on the right hand side can be evaluated using Eq Provided that 31 we can conclude that W ltwlt0gtiH1iw 31gtlt 2gttiH1iwlt0gtgt ltw 02lw532gt gt gt 5 E22 7 Elizxen 7 am Summary In cases of degeneracy it is necessary to work at least to second order in A to obtain WW correct to rst order If H1 does not lift the degeneracy between and ie then one must work to third order or higher PHY 6646 K lngersent TimeDependent Perturbation Theory The Photoelectric Effect 0 This handout mirrors the treatment ofthe photoelectric effect on Shankar pp 49975067 with two principal differences 1 The perturbing Hamiltonian is written H1E eER instead of HM emcA P 2 The system is assumed to occupy a cubic box of sides L7 whereas Shankar treats an in nite system We comment on the signi cance of these differences at the end The initial state is taken to belong to the innermost or K shell of a hydrogen like atom of effective nuclear charge Z6 with wave function ltrfz39 W 12Za032 exp7Zrao7 where 10 hzmez is the Bohr radius This state has energy 8139 7Z2622a0 7Zoz2ch27 a 62710 being the ne structure constant The characteristic size of the orbital is To aOZ hZamc We consider a monochromatic electromagnetic plane wave7 Er7 t E0 coskr 7wt The electric dipole approximation is valid provided that klro ltlt 17 or equivalently7 ha ltlt Zam02 We will consider frequencies in the window Za2m62 ltlt ha ltlt Z0077ch7 where not only can we make the dipole approximation7 but the nal state energy is suf ciently high that the nal state should be well described by a plane wave of the form ltrff L s2 expz39pfrh having an energy 8f pffz2m See Shankar p 500 and the end of this handout for discussion of this plane wave approximation In the dipole approximation7 we need to calculate the dipole matrix element r mmwr awwwrgmm a a A gtlt m d3r wwrE ZNac m fz39 5w 5pflt gt where A W 12ZLa032 The overlap integral is straightforward to evaluate see Shankar p 504 87TAZa0 239 3r 67ipfrh67Z IVao 39 m gt Ad Zao2 pfh22 Therefore r 3972 SWAZao lt74pfgt memmw Noting that Za02 pfh2 2m8f 7 80712 we nd 2m 47rAZh5 m Pflt zgt 2 Pf 1 r T a0m36f 7 813 o Fermi7s Golden Rule gives the scattering rate from to ff as 2 68f 7 81quot hw 27139 Riaf h 5 El 20rf 1