Fine Structure in Hydrogen - Relativistic Effects

LNPhysiqueAtomique2016 Fine structure in hydrogen - relativistic effects

Electron spin ; relativistic effects

• In a spectrum from H (or from an alkali), one ﬁnds that spectral lines appears in pairs. ⁃ take a Na spectrum as example:

• Moreover, the rays are slightly shifted in comparison with the non-relativistic theory

• The origins of these “new” effects: ⁃ Electron spin ⁃ Relativity

Relativistic Hamiltonian

• Recall the one-electron Hmiltonian: p2 Ze2 H = Hkin + V = 2m 4⇡"0 r

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p2 • So far, we have treated the term classically 2m

• For a more exact solution, this has to be replaced with a relativistic version

• This gives the “Dirac equation” ⁃ An analytical solution is possible, but very complex ⁃ Instead, we treat the problem with perturbation theory

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Perturbative treatment

• As a zero-order Hamiltonian, we take the non- relativistic version

• We the treat the relativistic corrections as perturbations 2 ~ 2 H = H + H0 = + V (~r)+H0 0 2mr

• The corrections to the energy levels:

E = H0 h 0| | 0i • It turns out that the relativistic corrections can be divided into three parts:

H0 = HSO + Hrel + HDarwin

⁃ HSO : Spin-orbit interaction ⁃ Hrel : Relativistic treatment of the kinetic energy ⁃ HDarwin : the Darwin term

⁃ This is consistent with the exact treatment

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Relativistic treatment of the kinetic energy

Hrel

• Classical kinetic energy : p2 E0 = kin 2m

• Relativistic kinetic energy : p2 p4 E = p2c2 + m2c4 mc2 = + ... kin 2m 8m3c2 p v 4 • First order correction (ignoring terms of order c or higher) : ⇣ ⌘ p4 ~4 H = = 4 rel 8m3c2 8m3c2 r

⁃ This does not depend on spin ⁃ It is diagonal in n and l

(Z↵)2 3 4 Erel,nl = En0 n2 4 l + 1 ✓ 2 ◆ e2 ⁃ where ↵ = 2 4⇡"0 c ~ ⁃ is the ﬁne-structure constant 1 ⁃ ↵ 137.036 ⇡

• Erel is more important for small n ⁃ (small n ⇒ small orbital radius ⁃ ⇒ high velocity)

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The Darwin term

HDarwin

• Very difﬁcult to explain …… ⁃ Related to the singularity at r =0

⇡~2 Ze2 H = (r) Darwin 2m2c2 4⇡" ✓ ◆

(Z↵)2 EDarwin,nl = En,0 n ; l =0 ) E =0 ; l =0 ( Darwin,nl 6

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Spin-orbit interaction

HSO

• Interaction between the orbital angular momentum and the spin

Spin

• An electron has a magnetic moment, which can be associated with a spin

⁃ Wave function : s,ms ⁃ Vectorial representation : sm | s i • The sm are eigenvectors to the operators: | s i 2 2 ⁃ S sms = s(s + 1)~ sms | i | i ⁃ Sz sms = ms~ sms | i | i • For a single electron, we have (always): 2 2 1 2 2 3~ p3~ - s = S = s(s + 1)~ = = 2 ) 4 2 ! 1 ⌦ ↵ ~ - m = S = s ±2 )hzi ± 2 ⁃ “spin-up” and “spin-down”

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• The total wave-function :

⁃ nlmlms ⁃ nlm m | l s i

Interaction between ~l and ~s

• The electron has a magnetic moment : µ~ ~s / • Consider a system of reference centered at the electron ⁃ ⇒ an orbiting proton (positive charge)

• An orbiting charge ⁃ ⇒ induced magnetic ﬁeld : B~ ~l /

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• Interaction between B~ and µ~ H = µ~ B~ ~l ~s L~ S~ SO · / · / · vectors operators | {z } | {z } • HSO is separable in radial and angular coordinates H = ⇠(r) L~ S~ SO · 1 1 dV ⁃ where ⇠(r)= 2m2c2 r dr

• For example, the hydrogenic potential: Ze2 1 Ze2 1 V = ⇠(r)= 2 2 3 4⇡"0 r ) 2m c 4⇡"0 r

• The energy shift due to the interaction is the expectation value of HSO

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Fine structure

• For the angular part of HSO : ⁃ we have to look for eigenvectors to L~ S~ h · i

• What about the vector : lml ms ? | i 2 ⁃ (eigenvector to the operators L , Lz , and Sz )

• This will NOT do, since L~ S,L~ =0 and · z 6 L~ S,S~ =0 h i · z 6 h i • ⇒ We need some other operator (and quantum number)

The total angular momentum

• We introduce : J~ = L~ + S~

J 2 = L2 +2L~ S~ + S2 1 · L~ S~ = J 2 L2 S2 ) · 2 • Consider the wave functions nljmj ( nljmj ) that 2 2 | i are eigenstates to H0 ,L ,J and Jz :

H0 nljmj = E0 nljmj 2 | i | 2 i L nljmj = l(l + 1)~ nljmj 2 | i 2| i J nljmj = j(j + 1)~ nljmj | i | i Jz nljmj = mj~ nljmj | i | i

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• For hydrogen, we have one single electron 1 s = 2 m = 1 ( s ± 2 1 j = l 2 , (l = 0) ± 1 6 ( j = 2 , (l = 0)

•j is a “good quantum number” ⁃ it makes the total Hamiltonian diagonal

• ml and ms are NOT good quantum numbers ⁃ due to the spin-orbit interaction, L~ and S~ will precess around each other ⁃ thus, their projections are not constant

• The sum, J~ , IS constant

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The Fine-structure energy

• Expectation value of the angular part of the hamiltonian: 1 1 L~ S~ = J 2 L2 S2 = ljm J 2 L2 S2 ljm · 2 2 j | | j D E ⌧ ~2⌦ ↵ = [j(j + 1) l(l + 1) s(s + 1)] 2 1 ⁃ with s = : 2 ~2 3 L~ S~ = j(j + 1) l(l + 1) · 2 4 D E 

H = ⇠(r) L~ S~ h SOi h i · 1 DZe2 E Z3 ~2 3 = j(j + 1) l(l + 1) 2m2c2 4⇡" a3n3 l(l + 1 )(l + 1) 2 4 0 0 2  1 3 = j(j + 1) l(l + 1) 2 4 

1 • The two possible values of ms ( ) gives two ±2 1 1 possible values for j ( l + and l ) 2 2

• ⇒ The energy level E0 is split into a doublet

+ 1 E = H H = l + SO SO SO 2 ✓ ◆ ⌦ ↵ ⌦ ↵

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Energy levels in hydrogen and spectroscopy

Spectroscopic notation

• There is a convention for how to annote quantum numbers ⁃ l-quantum numbers are described by a letter ⁃ The s-quantum number does not need description ⁃ The j-quantum number is given by its numerical value

• The combination of n and l (nl) is referred to as an “orbital”

• For a many-electron atom, the “electron conﬁguration” is the list of all orbitals ( n1l1,n2l2,... )

• Coding for l : l =0 s ! l =1 p ! l =2 d ! l =3 f ! l =4 g !

• For a multi-electron atom, total angular momenta have to be deﬁned.

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• These are given quantum numbers with capital letters: ~ ~ ~ L = l1 + l2 + ... quantum numbers L and ML ~ S = ~s1 + ~s2 + ... quantum numbers S and MS ~ ~ ~ J = l1 + ~s1 + l2 + ~s2 + ... quantum numbers J and MJ

• Coding for L : L =0 S ! L =1 P ! L =2 D ! L =3 F ! • Coding for S (multiplicity) : S =0 (2S + 1) = 1 singlet ! ! S =1/2 (2S + 1) = 2 doublet ! ! S =1 (2S + 1) = 3 triplet ! ! S =3/2 (2S + 1) = 4 quartet ! ! •J is given as its number ⁃ (for a multi-electron atom, the deﬁnition of J is ambiguous)

• The L and S together gives the “atomic term” 2S+1L ⁃ (examples : 3S , 2P , 4D , 1F ) ⁃ (not always a good description)

•J is the ﬁne structure level 2S+1 LJ 1 2 ⁃ (examples : S0 , P1/2 )

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Energy levels in hydrogen

• Ground state

n =1 l =0 ) ⁃ s =1/2 j =1/2 ) 1s 2S (1 level) ) 1/2 • Excites states l =0 - n =2 ,s=1/2 ) l =1 ⇢

2 2s j =1/2 2s S1/2 ) ) 2 j =1/2 2p P1/2 (3 levels) 2p ) 2 9 ) j =3/2 2p P3/2 ⇢ ) = ; l =0 - n =3 l =1 ,s=1/2 ) 8 < l =2

3s :3s 2S ) 1/2 3p 3p 2P and 3p 2P (5 levels) 1/2 3/2 9 3d ) 3d 2D and 3d 2D ) 3/2 5/2 = ;

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