CHAPTER 7 The Hydroge Ato 7.1 Applicatio of the Schrödiger Equatio to the Hydroge Ato 7.2 Solutio of the Schrödiger Equatio for Hydroge 7.3 Quatu Nubers 7.4 Magetic Effects o Atoic Spectra Noral Zeea Effect 7.5 Itrisic Spi 7.6 Eergy Levels ad Electro Probabilities By recogizig that the cheical ato is coposed of sigle separable electric quata, huaity has take a great step forward i the ivestigatio of the atural world. - Johaes Stark
7.4: Magetic Effects o Atoic Spectra The Noral Zeea Effect The Dutch physicist Pieter Zeea showed the spectral lies eitted by atos i a agetic field split ito ultiple eergy levels. It is called the Zeea effect. If lie is split ito three lies Noral Zeea effect If lie splits ito ore lies Aoalous Zeea effect (Chapter 8) Noral Zeea effect (A spectral lie is split ito three lies): Cosider the ato to behave like a sall aget. The curret loop has a agetic oet μ = IA ad the period T = 2πr / v. Thik of a electro as a orbitig circular curret loop of I = dq / dt : Magetic oet of Hydroge ato
The Noral Zeea Effect With o agetic field to alig the, poit i rado directios. However i a exteral field the dipole will experiece a torque which teds to alig it i the field. for ay orietatio the oet has a potetial eergy B Bohr ageto
The Noral Zeea Effect Depedig o the orietatio of the agetic oet (value of l ), the potetial of the orbitig electro i a agetic field is: Therefore each eergy level (l) of the orbitig electro which are (2 l +1) degeerate will be split by the applied agetic field. Whe a agetic field is applied, the 2p level of atoic hydroge is split ito three differet eergy states with eergy differece of E = μ B B l. l Eergy 1 E 0 + μ B B 0 E 0 1 E 0 μ B B
The Noral Zeea Effect 2p A trasitio fro 2p to 1s 1s 3 differet spectral lies due to photos eitted with slightly differet wavelegths fro field split levels.
Exaple 7.7 Deterie the value of the Bohr ageto, ad use this value to deterie the eergy differece betwee the l = 0 ad +1 copoets of the 2p state of atoic hydroge placed i a 2 Tesla agetic field. E E ev ev 2 2 1 / 2 13.6 / 4 3.4 Relatively speakig E is sall, however easily observed optically.
Ster Gerlach Experiet I 1920 s experiets were perfored to directly easure the space quatizatio of atos. I 1922 Ster ad Gerlach reported results of a experiet that clearly showed evidece for space quatizatio. Experiet was based o the idea that a ihoogeeous agetic field will exert a et force (i additio to a torque) o agetic dipole.
Ster Gerlach Experiet Ihoogeeous agetic field If it oves through a hoogeeous agetic field, the forces exerted o opposite eds of the dipole cacel each other out ad the trajectory of the particle is uaffected.
Ster-Gerlach Experiet A atoic bea of particles i the l = 1 state pass through a agetic field alog the z directio. The l = +1 state will be deflected dow, the l = 1 state up, ad the l = 0 state will be udeflected. If the space quatizatio were due to the agetic quatu uber l, l states is always odd (2l + 1) ad should have produced a odd uber of lies. Oe sigle lie (l = 0), 3 lies (l = 1), 5 lies (l =3), But, they observed oly two lies! The atos are deflected either up or dow! Why?
7.5: Itrisic Spi Clearly there was a proble with uber of lies observed i the Ster Gerlach experiet. I 1925, Sauel Goudsit ad George Uhlebeck i Hollad proposed that the electro ust have a itrisic agular oetu ad therefore a agetic oet. S s( s1) I order to explai experietal data, Goudsit ad Uhlebeck proposed that the electro ust have a itrisic spi quatu uber s = 1/2. Paul Ehrefest showed that the surface of the spiig electro should be ovig faster tha the speed of light! S : spi agular oetu s( s1) 3 2 Therefore this itrisic spi agular oetu ust be a purely quatu result.
Itrisic Spi (fourth quatu uber: s ) The spiig electro reacts siilarly to the orbitig electro i a agetic field. Spi will have quatities aalogous to L, L z, l, ad l. like L, the electro s spi ca ever be spiig with its agetic oet μ s exactly alog the z axis. The z copoet of the spi agular oetu: 1 Lsz s 2 s 1 2 The agetic spi quatu uber s has oly two values, sice (2s+1) = 2 depedig o the value of s, the spi will be either up or dow. Each electroic state i the ato ow described by four quatu ubers (, l, l, s )
Itrisic Spi The agetic oet due to itrisic spi is e e, or 2 B s S s S e B L, or 2. 2 The coefficiet of is 2μ B that of L / is B We ca defie the gyroagetic ratio (for l or s). The z copoet of. B Where g l = 1 ad g s = 2, the B, s g, s LS, B 2 B, B B s gs S S g L L We are ow i a positio to uderstad the results of the Ster Gerlach experiet that produced oly two distict lies. If electros are i a l = 0 state o splittig due to l. L But, there is space quatizatio due to the itrisic spi, s.
Itrisic Spi So sae arguet that was used previously for the orbitig electro ca ow be applied the agetic oet due to the itrisic spi. I a field the potetial eergy becoes: e e e VB sb SB sb B BB VBs, 2BB 2 e e VB B LB BBB VB, BB 2 2 Note: twice differece i s levels tha the differece i l levels. Special Topic: 21 c lie trasitio (page 260) Both proto ad electro have itrisic spi lowest eergy state is with the oppositely aliged. E betwee opposite ad parallel is 5.9 10-6 ev. correspods to a photo λ of 21 c (radio wave) widely used i Astrooy.
7.6: Eergy Levels ad Electro Probabilities For hydroge, the eergy level depeds o the priciple quatu uber. these eergies are predicted with great accuracy by the Bohr odel. I a agetic field the degeeracy is reoved. (hydroge ato) I ay-electro atos the degeeracy is also reoved either because of iteral B fields or because the average potetial of a electro due to ucleus plus electros is o-coulobic. geerally saller l states teds to lie at lower eergy for a give. For exaple, sodiu E(4s) < E(4p) < E(4d) < E(4f).
Selectio rules for radiative trasitio I groud state a ato caot eit radiatio. It ca absorb electroagetic radiatio, or gai eergy through ielastic bobardet by particles. Ca use the wave fuctios to calculate trasitio probabilities for electros chagig fro oe state to aother. Allowed trasitios: (for photo) Electros ca absorb or eit photos to chage states whe l = ±1. Chage i l iplies a L Z of ±ħ. Coservatio of agular oetu eas photo takes up this agular oetu. Forbidde trasitios:(for photo) Other trasitios possible but occur with uch saller probabilities whe l ±1. Selectio rule aythig 1 0, 1
( 참고 ) Radiative trasitios ad Selectio rules - Cocepts of Moder Physics, Beiser, Chapter 6.8 - What happes whe a electro goes fro oe state to aother? Eergy level E 에서 E 으로떨어질때의복사선의진동수 ()? E E h E h E 전자가한방향 (x-축 ) 으로움직이는 syste 을고려하면, 양자수 인 ( 에너지 E 인상태 ) 전자의시간에의존하는파동함수 : Ψ Ψ = ( 시간에무관한파동함수 ) x ( 주파수 ν = E /h 인시간의존함수 ) e ( ie / h) t 이러한전자의위치에대한기대치 <x> 는 x x dx x e ie / ie / t dx x dx
( 참고 ) x x dx 따라서, E 에너지상태에있는전자의 <x> 는 과 위치만의함수로정의되기때문에, <x> 값은시간에따라변하지않는다. 전자는진동하지않으므로어떠한복사 (radiatio) 도일어나지않는다 양자역학은특정한양자상태에머물러있는원자는복사하지않는다는것을예측하여주고있고관측결과와일치한다. Next, cosider a electro that shifts fro oe eergy state to aother. ( 다른 particle 과충돌혹은 radiatio 을흡수 / 방출하여 ) What is the frequecy of radiatio? 상태와 상태모두있을수있는전자의파동함수 Ψ. a b a a probability that the electro is i state b b probability that the electro is i state E E h
( 참고 ) a b a a + b b = a 2 + b 2 = 1 a= 1, b = 0 e i the groud state (E) a= 0, b = 1 e i the excited state (E) E E h e ( ie / h) t e ( ie / h) t 전자가위상태중어느곳에있든지복사 (radiatio) 을발생하지않음. 그러나, 에서 으로천이하는과정 (a 와 b 어느것도 0이아님 ) 에서는전자기파가발생된다. 중첩된파동함수의기대값 <x> 는 x x dx x a b a b dx x a 2 b a a b b 2 dx
( 참고 ) 2 ie/ t ie/ t x a x dx b a x e e dx ie/ t ie/ t 2 ab x e e dx b x dx e i e i cos i si cos i si cos E E t x b a a b dx i si E E t x b a a b dx
( 참고 ) 이결과의 Real part 는시간에따라다음과같이변한다. cos E E t cos E E 2 t cos 2 t h 따라서전자는삼각함수모양으로진동을하게되며, 그진동수는 E h 전자가 상태나 상태에머물러있을때전자의위치에대한기대값은상수이다. 진동하지않고복사도안일어난다 E 전자가두상태사이에서전이할때, 전자는진동수 로진동한다. 이러한전자는 electric dipole 같이동일한진동수 ( ) 를갖는 electroagetic wave 를복사한다. E E h
( 참고 ) Soe trasitios are ore likely to occur tha others Selectio rule 진동수 를알기위해서는시간의함수로서 a, b 의확률이나파동함수 과 를알필요는없다. 그러나, 어떤전이가일어날확률을계산하기위해서는이값들이필요하다. 여기상태에있는원자가복사하기위해필요한일반적인조건은적분 x dx 0 이어야하는데, 그것은복사강도가적분값에비례하기때문이다. 허용된전이 (allowed trasitio) 적분값 0 인경우금지된전이 (forbidde trasitio) 적분값 =0 인경우
( 참고 ) 수소원자의경우복사전이에관여하는처음상태와나중상태를규정하기위해 three quatu ubers 가필요하다. Iitial state :, l, l Fial state :, l, l Allowed trasitio u ', l, ', l ', l l 0 수소원자의,l,l 을알고있으므로 u=x, u=y, u=z 인경우위의값을구할수있다. 그결과 l 이 +1 과 1 만큼변하고 l 이바뀌지않거나 +1 또는 1 만큼바뀌는전위만일어난다. Selectio rules l = ±1 l = 0, ±1 주양자수 의변화는제한받지않는다. 선택규칙 l = +- 1 에의해허용되는전이를보여주는수소원자의에너지준위그림. 원자가복사를일으키기위해서 l = 1 이되어야한다는 selectio rule 은 photo 이처음과나중의각운동량차이에해당하는각운동량 (agular oetu) h 를갖고방출된다는것이다. 각운동량 h 를가지고있는광자란좌원편광혹은우원편광된전자기파라는의미임.
( 참고 ) Noral Zeea effect ad Selectio rules - Cocepts of Moder Pgysics, Chapter 6.10, Beiser- 자기장내에서는특정원자상태의에너지는 값뿐만아니라 l 의값에도관계한다. 복사전이의경우, l = 0, 1 로제한되기때문에서로다른 l 를가진갖는두상태에서생기는 spectru 선은 3 개의진동수만갖게된다. 진동수가 0 인 spectru 선이다음과같은진동수를갖는 3 개의성분으로분리됨. 1 2 3 0 0 0 B B B h B h 0 0 e 4 e 4 B B Noral Zeea effect
Probability Distributio Fuctios Ulike the plaetary odel of Bohr, i Schrödiger wave picture the positio of the electro i each state, defied by 4 quatu ubers (, l, l, s ), is spread over space ad is ot well defied. We ust use the wave fuctios to calculate the probability distributios of the electros The probability of fidig the electro i a differetial volue eleet d is: i spherical polar coordiates: ϕ depedece: Ig(ϕ)I 2 cotais factor (e ilϕ ) e ilϕ = 1 thus o ϕ depedece i probability thus wave fuctios are syetric about z-axis.
Probability Distributio Fuctios If we are oly iterested i the radial probability distributios of the electro, ad, if the fuctios f(θ) ad g(ϕ) have bee properly oralized the itegratio over all values of ϴ, ad ϕ yields: Therefore, : Radial Probability Note: the radial probability desity P(r) = r 2 R(r) 2 depeds oly o ad l.
Probability Distributio Fuctios R () r 2 P () r r R 2 Observatios: < r > depeds o or eergy, ad is idepedet of l. < r > 2 (sae as Bohr). ost probable radii for 1s ad 2p agree with Bohr result (a 0 ad 4a 0 respectively). true for wave fuctios with axiu l R(r) ad P(r) for the lowest-lyig states of the hydroge ato
Probability Distributio Fuctios R () r 2 P () r r R 2
Probability Distributio Fuctios 2 P () r r R 2 Exaple 7.11 Fid the ost probable radius for the electro i a hydroge ato i the 1s state. To fid the axiu we set the derivative of the radial probability equal to zero. Exaple 7.12: Calculate the average orbital radius of 1s. < r > = (3/2) a 0
Probability Distributio Fuctios 2 P () r r R 2 Exaple 7.13 What is the probability of a electro i the 1s state of the hydroge ato beig at a radius greater tha the Bohr radius?
The electro cloud i the hydroge ato
Probability Distributio Fuctios (, r, ),,