Perturbation Theory, Zeeman Effect, Stark Effect

Transcription

1 Chapter 8 Perturbatio Theory, Zeea Effect, Stark Effect Ufortuately, apart fro a few siple exaples, the Schrödiger equatio is geerally ot exactly solvable ad we therefore have to rely upo approxiative ethods to deal with ore realistic situatios. Such ethods iclude perturbatio theory, the variatioal ethod ad the WKB -approxiatio. I our Scriptu we, however, just cope with perturbatio theory i its siplest versio. It is a systeatic procedure for obtaiig approxiate solutios to the uperturbed proble which is assued to be kow exactly. 8. Tie Idepedet Perturbatio Theory The ethod of perturbatio theory is that we defor slightly perturb a kow Hailtoia H by a ew Hailtoia H I, soe potetial resposible for a iteractio of the syste, ad try to solve the Schrödiger equatio approxiately sice, i geeral, we will be uable to solve it exactly. Thus we start with a Hailtoia of the followig for H = H + λ H I where λ [, ], (8.) where the Hailtoia H is perturbed by a saller ter, the iteractio H I with λ sall. The uperturbed Hailtoia is assued to be solved ad has well-kow eigefuctios ad eigevalues, i.e. H = E (), (8.2) where the eigefuctios are chose to be oralized = δ. (8.3) The superscript deotes the eigefuctios ad eigevalues of the uperturbed syste H, ad we furtherore require the uperturbed eigevalues to be o-degeerate, aed after Wetzel, Kraers ad Brilloui. E () E (), (8.4) 5

2 52CHAPTER 8. PERTURBATION THEORY, ZEEMAN EFFECT, STARK EFFECT otherwise we would use a differet ethod leadig to the so-called degeerate perturbatio theory. What we are ow goig to ivestigate are the eigevalues E ad eigefuctios of the total Hailtoia H H = E. (8.5) The basic idea of perturbatio theory the is to expad the eigevalues ad eigefuctios i a power series, without takig care of the correspodig covergece properties. I ay cases, cosiderig just the first few ters of the power series gives us reasoable results, eve if the whole power series ca be show to be diverget. Let us therefore start with a power series i the paraeter λ, which is a good approxiatio for sall λ E = E () + λ E () + λ 2 E (2) + (8.6) = + λ + λ 2 2 +, (8.7) where we have coefficiet fuctios of labda, the uperturbed ter, first ad secod order perturbative correctios ad so o. Isertig these expasios ito the Schrödiger equatio (8.5) together with Hailtoia (8.), we get = (H + λ H I ) ( + λ + λ ) = ( ) E () + λ E () + λ 2 E (2) + ( + λ + λ ). (8.8) Equatig coefficiets we obtai the followig relatios: λ : H = E () (8.9) λ : H + H I = E () + E () (8.) λ 2 : H 2 + H I = E () 2 + E () + E (2). (8.) Additioally, it is coveiet to oralize the followig trasitio aplitude for the perturbed eergy eigestate = (8.2) 8.. First Order Correctios λ + λ = (8.3) = 2 = =. (8.4) To derive the first order correctios we ultiply the first order coefficiet equatio (Eq. 8.) with fro the left ad let the heritia, uperturbed Hailtoia H = H act to the left o E () } {{ } + HI = E () } {{ } + E () } {{ }. (8.5) This leads us directly to the followig fudaetal result, which is already a sufficiet correctio for ay cases to reproduce physical observatios:

3 8.. TIME INDEPENDENT PERTURBATION THEORY 53 Theore 8. The perturbative eergy correctios of first order are give by the expectatio value of the perturbig Hailtoia i the uperturbed states E () = HI Although we ow have the eergy correctio we are iterested i, we are curious as to how the state itself is perturbed. We therefore expad the first order expasio state ito a coplete orthooral syste { } =, (8.6) where all the ters for = vaish due to our choice of oralizatio (Eq. 8.4). We are the iterested i the expasio coefficiet, which we calculate by usig agai Eq. (8.). This tie, however, we ultiply the equatio with E () + HI = E () + E (). (8.7) Though the last ter o the right side of Eq. (8.7) would be ozero if =, the state expasio excludes this case i the suatio ad therefore we ca igore this ter, which allows us to express the expasio coefficiet as = H I E () E (), (8.8) where the atrix eleet H I has to be calculated. So the iportat result for the first order state correctio is: Theore 8.2 The perturbative state correctios of first order are give by the superpositio of all uperturbed states with eergies differet fro the cosidered state = H I E () E () Reark: Due to the eergy deoiator the adjacet eergy levels cotribute i this approxiatio stroger to the correctios of the state (assuig early costat atrix eleets) tha the ore reote oes, what is ituitively expected.

4 54CHAPTER 8. PERTURBATION THEORY, ZEEMAN EFFECT, STARK EFFECT 8..2 Secod Order Correctios To fid the correctios of secod order, we start siilar as i the last subsectio, by this tie ultiplyig Eq. (8.) by E () 2 + HI = E () 2 + E () + E (2). }{{}}{{}}{{}}{{} (8.9) Further isertig our result of Sectio 8.. (Theore 8.2), we get E (2) = HI H I E () E (), (8.2) the we iediately fid the result for the secod order eergy correctio as: Theore 8.3 The perturbative eergy correctios of secod order are deteried by all uperturbed states with eergy differet fro the cosidered state E (2) = H I 2 E () E () Reark I: Agai, the adjacet eergy levels cotribute to the ajority of the correctio if the atrix eleets are assued to be of approxiately the sae order. Reark II: For the groud state the eergy correctio of secod order is always egative, i.e. E (2) <, sice E () < E (). 8.2 Zeea Effect The eergy levels of particles carryig a agetic dipole oet, e.g. the H-ato, i a exterior (hoogeeous) agetic field are further split up, i.e., the degeeracy of the eergy eigevalues is reoved. The Hailtoia of the electro i the H-ato therefore has a additioal ter, the Zeea ter H Z, perturbig the already kow ter H Coul. which cotais the Coulob potetial (Eq. 6.95). We thus have the total Hailtoia H = H Coul. + H Z, (8.2) where the Zeea ter is the scalar product of the agetic dipole oet µ ad the exterior agetic field B, see Eq. (7.4) H Z = µ B = q ( ) L + g S B. (8.22) 2 c

5 8.3. STARK EFFECT 55 We ow assue without loss of geerality that the agetic field is orieted alog the z-directio, where it has the stregth B ext., ad isert the electro ass e, charge e = e ad g-factor g e = 2 H Z = e 2 e c ( L z + 2 S z ) B ext.. (8.23) We further assue that the agetic field is sufficietly strog, such that the orbital agular oetu L ad the spi S decouple ad the uperturbed states are product states of the for, l, l ad, l, l, (8.24) which is called the Pasche Back Effect. The (first order) eergy correctios E to the pure Coulob eergy are the give by Theore 8., where we iediately see that E (), l, l, s = e 2 e c ( l + 2 }{{} s ) B ext.. (8.25) ± Thus every, util ow degeerate, l-level is split up by the eergy correctio ter, depedig o the spi orietatio. Whe recallig the eergy levels of the Coulob potetial (Eq. 6.4) E, Coul. l = E I 3, 6 ev =, (8.26) 2 2 ad Bohr s ageto µ B µ B = e 2 e c = 5, ev / Tesla, (8.27) we see that the Zeea eergy ter is a sall correctio with respect to the Coulob eergy, eve for very strog agetic fields (B ext. Tesla). 8.3 Stark Effect The Stark effect is the electric aalogue to the Zeea effect, i.e., a particle carryig a electric dipole oet, like the H-ato, will get a splittig of its eergy levels whe subjected to a exterior electric field. The Hailtoia of the H-ato thus has (aother) additioal ter, the Stark ter H Stark, which is perturbig the Coulob Hailtoia H Stark = E µ el = q }{{} e E x = e E z, (8.28) where E deotes the exteral electric field which, without loss of geerality, we choose alog the z-directio, ad µ el is the electrical dipole oet. For the siple case of a electro of charge e placed at a distace x of the oppositely charged proto, restig at the origi, the electric dipole oet reduces to µ el = e x.

6 56CHAPTER 8. PERTURBATION THEORY, ZEEMAN EFFECT, STARK EFFECT To calculate the eergy correctios of first (Theore 8.) ad secod (Theore 8.3) order, we eed to cosider expectatio values or scalar products of the Stark ter i the Coulob states of the for, l, l. This is, however, ore coplicated tha the situatio we studied before. To ake the proble ore precise, let us calculate the followig scalar product, where we already kow fro Eq. (6.5) that L z ad z coute, l, l [ L z, z ], l, l = ( }{{} l l ), l, l z, l, l =. (8.29) = Sice the coutator of L z ad z is zero we ca coclude that the states, l, l ad z, l, l ust be orthogoal for l l, ad sice we are oly iterested i ozero atrix eleets we choose l l. (8.3) At the sae tie, however, for the scalar product to be ozero, we have to require that the aziuthal quatu ubers be uequal, i.e. l l. The reaso for this is the parity of the z operator. We reeber fro Theore 4.2 that a basis of states ca be chose for a syetric potetial, such as the Coulob potetial (Eq. (6.95)), that cosists etirely of eve ad odd fuctios, which i this case are the Legedre polyoials P l (cos θ). These are polyoials of order l i cos θ = z ad thus alterately eve ad odd r fuctios. Therefore, if the fuctios ψ, l, l ad ψ, l, l have the sae aziuthal quatu uber, i.e. parity, their product has parity +. The z operator however chages its sig uder a spatial iversio ad the itegratio of the scalar product vaishes, because We ca the assue that +a a dx f(x) g(x) }{{}}{{} eve odd. (8.3) l = l ±, (8.32) which is a reasoable choice sice the electric dipole iteracts with the electric field by exchagig photos of spi, so the agular oetu of the electro ust decrease or icrease by. The Stark effect will therefore oly occur for atrix eleets of the for, l, l z, (l ± ), l =. (8.33) Let us ow cosider the eergy correctio for the groud state of the hydroge ato, the first order correctio is E (),, = e E,, z,, =. (8.34) It vaishes due to the requireet of Eq. (8.32) ad we thus see that there is o liear Stark effect for the hydroge groud state. The effect of secod order is of the for E (2),, = e 2 E 2,, z,, 2, (8.35) E () E () = 2

7 8.3. STARK EFFECT 57 where the uperturbed eergy levels E () are give by Eq. (6.5). This expressio ca be calculated usig perturbative ethods, which leads to the result E (2),, = 9 4 where r B is the Bohr radius (Eq. (6.97)). 2 e e 2 E 2 = 9 4 r B E 2, (8.36) The first excited state, i cotrast to the groud state, is four-fold degeerate, i.e. there are 4 states 2,,, 2,,, 2,, ad 2,,, that belog to the pricipal quatu uber = 2. I this case oe has to cosider degeerate perturbatio theory, which leads to calculatig the eigevalues of a o-diagoal atrix. Because of the requireet of Eq. (8.3) we oly cosider eleets with equal agetic quatu uber, which is oly the case for the states 2,, ad 2,,. It leaves us a 2 2 atrix to diagoalize. The secod requireet (Eq. (8.32)) rids us of the diagoal eleets of the atrix. The reaiig ozero off-diagoal eleets are 2,, z 2,, = 2,, z 2,,. (8.37) The ozero eigevalues ca thus be easily calculated ad we coclude that the excited state has a liear Stark effect correctio with the result E () 2, l, l = ± e r B E. (8.38) Further splittig of the excited hydroge eergy levels ca be see i Fig. 8..

8 58CHAPTER 8. PERTURBATION THEORY, ZEEMAN EFFECT, STARK EFFECT Figure 8.: Stark Effect i Hydroge: The util the degeerate excited eergy levels are split up if a exterior electric field is applied. Figure fro: splittig.pg