Lecture 10 Highlights We started with the second order corrections to the perturbed Schrödinger equation: , (1) nnnEψψ=Ηsolved assuming: ...2210+++=nnnnψλλψψψ (2) (3) ...2210+++=nnnnEEEEλλand yielding (to second order): (4) 0211201202':nnnnnnnnEEEψψψψψλ++=Η+ΗThe second-order equation can be solved using the fact that 1nψand 2nψcan each be expressed as a linear combination of all the eigenfunctions of 0Η(a postulate of QM) as, ∑≠=nkknkna01ψψ ∑=lll02ψψnnb (5) where the are known from the solution of the first-order equation in the last lecture, but the are unknown at this point. Putting (5) into (4) and exploiting orthonormality (i.e. multiply both sides by nkalnb*0jψand integrating over all space) yields (for the case nj=): ∑∫≠−Η=nkknnknEErdE00230*02'ψψ (6) This represents the second order correction to the energy. It is often necessary to calculate this because the first-order energy correction is sometimes zero. This result again assumes that the energy eigenvalues are non-degenerate. As an example of first-order perturbation theory we considered the relativistic correction to the kinetic energy operator. Following the discussion in Griffiths pages 267-270 we found a relativistic correction to the kinetic energy operator as: 234282 cmpmpT −= The new Schrödinger equation for the Hydrogen atom can now be written as: ψψE=Η , with , and '0Η+Η=Ηremp022042πε−=Η is the original un-perturbed Hydrogen atom Hamiltonian, and 2348'cmp−=Η is the perturbation. We evaluate the change in energy to first order using the result derived in the last lecture: , rdEnnn30*01'ψψ∫∫∫Η=where 0nψare the unperturbed Hydrogen atom wavefunctions, and now represents the list of H-atom quantum numbers . Evaluating the expectation value integral as in Griffiths yields the following result: nmn ,,l 1⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−+−= 321422201,llnnEEnnα where the subscripts are now the principle quantum number n and angular momentum quantum number of the Hydrogen atom, and . We have also introduced a new and very important dimension-less parameter called the fine structure constant l20/eV 6.13 nEn−=α. This is a combination of fundamental constants from electrodynamics, quantum mechanics and relativity: 036.1371402≅≡cehπεα. Note that the correction to the energy of the Hydrogen atom due to relativistic effects is on the scale of , which is roughly on the order of eV, as compared to the ground state energy of order 10 eV. Also note that the dependence of the first-order corrected energy will lift some of the degeneracies of the un-perturbed hydrogen atom, and this will give rise to “fine structure” in the radiation emission spectrum of the atom. In other words some of the H-atom spectral lines will now be split into multiple lines (because of the dependence of ) with an energy splitting on order eV. Such effects are visible in a spectrometer as “fine structure splitting” of the spectral lines. 02nEα310−ll1,lnE310−
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