Lecture 3: Helium Readings: Foot Chapter 3

Transcription

1 Lecture 3: Helium Readings: Foot Chapter 3 Last Week: the hydrogen atom, eigenstate wave functions, and the gross and fine energy structure for hydrogen-like single-electron atoms E n Z n = hcr Zα / µ c ( µ m) = n Rydberg formula + fine-structure terms coming in at higher powers of αα This week: multi-electron atoms (helium, alkali metal atoms),

2 What we ll learn from helium How to deal with electron-electron interactions (screening) How to deal with quantum statistics -restrictions on filling -exchange terms Consequence of transition selection rules for spin

3 Helium gross structure and eigenstates H = + + H H H int H = Ze Ze e + m 4πε r m 4πε r 4πε r

4 Helium gross structure and eigenstates H = + + H H H int H = Ze Ze e + m 4πε r m 4πε r 4πε r For He + : th order approx: just ignore H int E gs n Z = = ( 3.6 ev) = 54.4 ev n = (n,l,m l ) = (,,) (,,-) (,,) (,,) n = (n,l,m l ) = (,,)

5 Helium gross structure and eigenstates H = + + H H H int H = Ze Ze e + m 4πε r m 4πε r 4πε r For He + : th order approx: just ignore H int E gs n Z = = ( 3.6 ev) = 54.4 ev Pauli exclusion: electrons can t have all the same quantum # s n = (n,l,m l ) = (,,) (,,-) (,,) (,,) ψ ( ) ( ) ( ) gs, spatial = ψ s r ψ s r s m s {-/, /} n = (n,l,m l ) = (,,)

6 Helium gross structure and eigenstates H = + + H H H int H = Ze Ze e + m 4πε r m 4πε r 4πε r For He + : th order approx: just ignore H int E gs n Z = = ( ) ψ gs, spatial = s expt E gs ( 3.6 ev) ( ) E gs 79 ev 9 ev = 54.4 ev m s {-/, /} n = (n,l,m l ) = (,,) (,,-) (,,) (,,) n = (n,l,m l ) = (,,)

7 Helium gross structure Now let s estimate, to first order, the effect of interactions ψ E ( ) Z = Z = Zr / a Zr / a ( ) ( ) gs, spatial = H = s int = = R 4π s e 4πε r r R s r 34 ev The derivation of this result is basically HW question #3a so won t go through it in detail = Z πa The basic idea is to consider the effective potential felt by electron # due to Coulomb interactions with average distribution of electron # s charge (and vice versa) 3 3 e e This is not a great estimate it s based on first-order perturbation theory and assumes that the eigenstates don t change much due to H int. However, ΔE is on the order of E, so H int is a large perturbation. >> HW question 3b asks you to use a variational approach to minimize the energy

8 Multi-electron wave functions - symmetry Electrons, being fermions, must* have a total wave function that is antisymmetric under the exchange of two particles ψ ( r r,, r ) = ψ ( r, r,, ), N r N * in 3 dimensions Let s restrict to just two particles, and label states by all their relevant quantum numbers Q = (n,l,m l,m s ) spatial quantum numbers: n, l, m l for our hydrogen-like orbitals spin quantum number: each electron can have m s = ±/, which we ll write as and

9 Multi-electron wave functions - symmetry The solution ( r, r ) ψ Q ( r ) ψ ( r ) ψ = is not allowed for Q = Q Q We can exclude this possibility by instead taking the superposition state ψ ( r, r ) = ψ ( r ) ψ ( r ) ψ ( r ) ψ ( r ) ( ) Q Q Q Q which vanishes for Q = Q naturally gives us ψ ( r, r ) = ψ ( r r ),

10 Multi-electron wave functions - symmetry The g.s. spatial wave function (s ) for helium is symmetric space ( r r ) ( r ) ψ ( ) ψ =ψ, s s r The spin part of the g.s. wave function must be antisymmetric ψ spin ( ) ( r, r ) = ψ ( r ) ψ ( r ) ψ ( r ) ψ ( r ) singlet state with total spin S = and m S = (parahelium)

11 Multi-electron wave functions - symmetry The two-electron spin wave function can also be symmetric, and there are three possibilities: ψ spin ( ) = r, r { m ( r ) ( ) ψ ψ r ( r ) ψ ( r ) ψ ( r ) ( r ) ( ψ ) + ψ ( r ) ( ) ψ ψ r m S = + S = m S = - These are the triplet states with total spin S = These configs require an antisymmetric spatial wave function (orthohelium)

12 Excited states Let s consider the most relevant class of excited states where one of the electrons remains in the s orbital. Excited states can be both singlet and triplet spin configurations, and thus both symmetric and antisymmetric spatial configurations At this gross structure level, the energies are not explicitly tied to the spin of the electrons, but we ll find that the symmetric & antisymmetric spatial wave functions can have different energies (Hund s rule #)

13 Which state do you expect to have a lower energy, P or 3 P? A) P B) 3 P C) the same energy

14 Which state do you expect to have a lower energy, P or 3 P? A) P B) 3 P This triplet, symmetric spin state has an antisymmetric spatial wave function with reduced screening it is more tightly bound C) the same energy

15 What is (approximately) the total energy of a state with electrons in the s and 3p orbitals?

16 What is (approximately) the total energy of a state with electrons in the s and 3p orbitals? Electron-electron interactions are not that important for these far-separated orbitals (also recall that the 3p orbital goes to zero at the nucleus). For the s electron, we can ignore shielding by the 3p electron, and so it s energy is roughly (Z=/n=) 3.6 ev = ev The 3p electron orbital experiences almost full shielding by the s electron, effectively seeing a hydrogen-like potential with Z eff =, and has an energy of roughly (Z eff =/n=3) 3.6 ev = -.5 ev The total energy should be roughly ev

17 a bit on notation / selection rules Spectroscopic notation: (n,l) nl J (shorthand for n S+ L J ) see Foot.8,.,.35 Spin term is typically omitted for hydrogen and alkalis, as S = / for the single (unpaired, valence) electron, so all states have S+ = For electric dipole transitions (basically assuming E-field of absorbed/emitted light is uniform over the size of the atom) H H d Δl, Δj from cons. of angular momentum Δl from symmetry of dipole operator d = d E Δn: no limit = ( er ) E Δl = ± iωt Δm l =, ± H d ee Δj =, ± Δm j =, ± always: no j = j = from conservation of angular momentum ( r ˆ ε ) e

18 Dipole-forbidden transitions One consequence: direct s s, p p, etc. transitions are forbidden. S / state is metastable (can t decay directly via stimulated emission) Lamb shift experiment relied on long lifetime of S / ττ. seconds S / can undergo two-photon decay [through intermediate state(s)]

19 Metastable helium The net spin of multi-electron atoms like helium will give a new important selection rule, ΔS = (dipole operator doesn t act on spin) 3 S S transition is doubly forbidden (l = l =, ΔS = ) metastable helium lives for over hours! long enough to do some cool science Long lifetimes also for of neon (5 s), argon (38 s), krypton (39 s), xenon (43 s)

20 Laser cooling of noble gases P ~.3 ev ~58 nm Zeilinger group Also, laser-cooling and trapping of neon, argon, krypton, xenon

21 Metastable helium: Atom optics with nano-grenades Lot s of internal energy left to give up Olivier Sirjean, Orsay group From Physics Today article Can kick out electrons / be used for lithography etched gratings in gold

22 Metastable helium: Atom optics with nano-grenades high-efficiency single atom detection Palaiseau group Vienna group

23 Bosons and fermions Palaiseau group

24 ANU group Other cool stuff