Hydrogen (atoms, molecules) in external fields. Static electric and magnetic fields Oscyllating electromagnetic fields

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1 Hydroge (atoms, molecules) i exteral fields Static electric ad magetic fields Oscyllatig electromagetic fields

2 Everythig said up to ow has to be modified more or less strogly if we cosider atoms (ad ios) which are ot isolated, but iflueced by a exteral electromagetic field. For low-lyig boud states of a atom the ifluece of exteral fields ca ofte be satisfactorily accouted for with perturbative methods, but this is o loger possible for highly excited states ad/or very strog fields, i which case itricate ad physically iterestig effects occur,eve i the simple hydroge atom. The study of atoms (ad molecules) i strog exteral fields has bee a topic of cosiderable iterest.

3 Geeral itroductio I this sectio we cosider a classical electromagetic field described by the scalar potetial Φ(r, t) ad the vector potetial A(r, t). 2 [ p+ ( e / c) A( r, t)] H = eϕ( r, t) + V 2μ A importat cosequece of exteral fields is, that the Hamiltoia is i geeral o loger rotatioally ivariat, so that its eigestates are t simultaeously eigestates of agular mometum. For spatially homogeeous fields ad the appropriate represetatio of Hamiltoia it remais ivariat uder rotatios aroud a axis parallel to the directio of the field, so that the compoet of total agular mometum i the directio of the field remais a costat of motio. For a electro i a potetial V (r) which is ot radially symmetric, but ivariat uder rotatios aroud the z-axis, say, we ca at least reduce the three-dimesioal problem to a twodimesioal problem by trasformig to cylidrical coordiates ρ, z, φ

4 With the asatz:

5 Static homogeeous electric field We describe a static homogeeous electric field E, which is take to poit i the directio of the z-axis, by a time-idepedet scalar potetial φ (vector potetial vaishes) The Hamiltoia the has the followig special form: 2 p H = + V + 2μ eze z The shifts i the eergy eigevalues caused by the cotributio of the field are give i time-idepedet perturbatio theory.

6 ΔE 1 = ee z ψ zψ where ψ zψ = ψ * zψ dτ where ψ are the eigestates of the uperturbed (E z = 0) Hamiltoia. These eigestates are usually eigestates of the electro parity operator so that the expectatio value of the operator z, which chages the parity, vaish. I the uusual case that a eigevalue of the uperturbed Hamiltoia is degeerate ad has eigestates of differet parity, we already obtai ovaishig eergy shifts i first order, ad this is called the liear Stark effect. The first-order eergy shifts are calculated by diagoalizig the perturbig operator ee z z i the subspace of the eigestates with the degeerate (uperturbed) eergy.

7 From the aalysis of the secod order perturbatio oe obtais: ΔE (2) = ( ee z ) 2 ΔE ψ E (2) zψ m 2 Em α d = 2 E where 2 z m α d is dipole polarizability E ad Em are the eigevalues of the uperturbed Hamiltoia. The eergy shifts depeds quadratically o the stregth Ez of the electric field ad are kow uder the ame quadratic Stark effect.

8 For = 2 there is a o-vaishig matrix elemet betwee the l = 0 ad l = 1 states with m = 0. The two further l = 1 states with azimuthal quatum umbers m = +1 ad m = 1 are uaffected by the liear Stark effect. Figure shows the splittig of the = 2 term i the hydroge atom due to the liear Stark effect. For compariso Figure shows the eergy shift of the = 1 level due to the quadratic Stark effect.

9 Importat extras 1. The eergy shifts i quadratic Stark effect are closely coected with the dipole polarizability of the atom i a electric field. 2. The wave fuctios are o loger eigefuctios of the electro parity, ad they have a dipole momet iduced by the exteral field ad poitig i the directio of the field (the z-directio). Field ioizatio: -importat i spectroscopy of Rydberg states -large field geerated by the lasers ca reduce effective eergy for the atom (molecule) ioizatio.

10 Atoms i a static, homogeeous magetic field A static homogeeous magetic field poitig i z directio ca be described i the symmetric gauge by a vector potetial I this gauge the Hamiltoia (3.225) keeps its axial symmetry aroud the z-axis ad has the followig special form: H = 2 p + V + 2μ e e 2 2 Bz Lz + Bz ( x + 2μ 8μ 2 y 2 ) μ is reduced mass We eglect the term i the Hamiltoia (3.249) which is quadratic i the field stregth Bz, ad cosider oly liear compoet.

11 The hamiltoia cotais compoet (Zeema iteractio Hamiltoia) describig the iteractio of the magetic field with magetic momet geerated by orbital agular mometum with exteral magetic field. Assumig that the iteractio is small eough, first order perturbatio theory gives the Zeema eergy: ormal Zeema effect Eigestates of the uperturbed (field-free) Hamiltoia, i which effects of spi-orbit couplig are egligible ad i which the total spi vaishes, i.e. i which the orbital agular mometum equals the total agular mometum, remai eigestates of the Hamiltoia i the presece of the magetic field, but the degeeracy i the quatum umber M L is lifted.

13 We geerally caot eglect the cotributios of the spi to the eergy shifts i a magetic field. The most importat cotributio comes from the magetic momets due to the spis of the electros. The iteractio of these spi momets with a magetic field is obtaied most directly if we itroduce the field ito the Dirac equatio ad perform the trasitio to the o-relativistic Schr odiger equatio. To first order we obtai the followig Hamiltoia for a free electro i a exteral magetic field: H = 2 p + V 2μ + e Bz ( Lz + 2μ 2 S) The iteractio of a atom with a magetic field is thus give to first order i the field stregth by a cotributio.

14 The magetic momet ow is o loger simply proportioal to the total agular mometum ˆJ = ˆL + ˆS, which meas that there is o costat gyromagetic ratio. The splittig of the eergy levels i the magetic field ow depeds ot oly o the field stregth ad the azimuthal quatum umber as i the ormal Zeema effect; for this reaso the more geeral case, i which the spi of the atomic electros plays a role, is called aomalous Zeema effect. ( μ J =g J γj )

15 As the stregth of the magetic field icreases, the iteractio with the field becomes stroger tha the effects of spi-orbit couplig. It is the sesible to first calculate the atomic states without spi-orbit couplig ad to classify them accordig to the quatum umbers of the z- compoets of the total orbital agular mometum ad the total spi: ΨL,S,ML,MS. The eergy shifts due to the iteractio with the magetic field (3.254) are the without ay further perturbative assumptios simply This is the Pasche-Back-Effekt.

16 Vector model L S J S L Bz Bz Uder spi-orbit domiace Pasche-Beck effect The spi ad orbital agular mometum are decoupled

17 Schematic illustratio of level splittig i a magetic field for the example of a 2P 1/2 ad a 2P 3/2 multiplet, which are separated by a spi-orbit splittig ΔE 0 i the field-free case. If the product of field stregth B ad mageto is smaller tha ΔE 0 we obtai the level splittig of the aomalous Zeema effect for μ B B>ΔE 0 we eter the regio of the Pasche-Back effect

18 Atoms i a Oscillatig Electric Field The theory of the iteractio betwee a atom ad the electromagetic field describes the resoat absorptio ad emissio of photos betwee statioary eigestates of the field-free atom. But a atom is also iflueced by a (moochromatic) electromagetic field if its frequecy does t happe to match the eergy of a allowed trasitio. For small itesities oe obtai splittig ad frequecy-depedet shifts of eergy levels (ac Stark shift, frequecy depedat polarizability); for sufficietly high itesities as are easily realized by moder laser techology,multiphoto processes (excitatio, ioizatio) play a importat role.

19 Extras Molecular Aligmet ad Orietatio: From Laser-Iduced Mechaisms to Optimal Cotrol Laser-iduced molecular aligmet ad orietatio are challegig cotrol issues with a wide rage of applicatios, extedig from chemical reactivity to aoscale desig. They address molecular maipulatio, ivolvig exteral agular degrees of freedom, aimig at a parallel positioig of the molecular axis with respect to the laser polarizatio vector (aligmet ) or, eve more demadig, with a give directio (orietatio).

20 E.g., HCN H=H 0 + H rad where The dyamics of the molecular system are the obtaied by solvig umerically the time-depedet Schrödiger equatio:

Lecture 25 (Dec. 6, 2017)

Lecture 25 (Dec. 6, 2017) Lecture 5 8.31 Quatum Theory I, Fall 017 106 Lecture 5 (Dec. 6, 017) 5.1 Degeerate Perturbatio Theory Previously, whe discussig perturbatio theory, we restricted ourselves to the case where the uperturbed