Please enable JavaScript.

Coggle requires JavaScript to display documents.

AMO (Fine structure (∆E=-(aZ)^2/n*(3/4n-1/(j+1/2)) E_n (darwin-term :red…

- - - - Quantization of E-M-fields
        wave equation: \(\vec{\nabla}^2\vec A-\frac{1}{c^2}\frac{\partial^2\vec{A}}{\partial t^2}=0\)
        solution: superposition of modes
        a mode \(\vec A(\vec r,t)=\frac{1}{\sqrt{V}}[c_0\vec \epsilon e^{i(\vec k \vec r - \omega t)}+c.c.]\)
        
        energy of mode
        \(E=\int \frac{1}{2}(\epsilon_0\vec E^2+\frac{1}{\mu_0}\vec B^2)dV)\)
        use \(A, E=-\dot A, B=\nabla\times A\)
        \(E=2\epsilon_0 \omega^2|c(t)|^2\)
        
        Quantization of classical harm oscillator
        \(H=\frac{p^2}{2m}+\frac{m\omega^2}{2}q^2\)
        \(\Rightarrow H=\hbar\omega(\hat P^2+\hat Q^2)\)
        
        raising/lowering op
        \(\hat a:=\hat Q+i\hat P\)
        \(\hat a\dagger:=\hat Q-i\hat P\)
        \(\hat a\dagger |n\rangle=\sqrt{n+1}|n+1\rangle\)
        \(\hat a |n\rangle=\sqrt{n}|n-1\rangle\)
        
        \(\vec E (\vec r,t) =\sqrt{\frac{\hbar}{2\epsilon_0\omega V}}\omega[i\hat a\vec \epsilon e^{i(\vec k \vec r - \omega t)}+h.c.]\)
- - - - quantum defect (empirical description of spectrum)
        weakest bound e-: E=-R_y*(1/(n-∂_l))^2
        =>E&l: low l => close to nucleus :red_flag:
    - - Russel-Saunders Coupling (ignore S-O term)
        ES are ES of L, S
        obey angular momentum coupling (Clebsch-Gordons)
        and Pauli-principle
      - ground state?
        Hunds rules:
        
        max S -> min E (e same spin direction->less spatial overlap->less repulsionl
        
        if max S: max L -> min E
        (same l => e meet less often)
        
        outer shell not full => min J
        otherwise max J
      - j-j coupling
        if S-O > residual
        large atoms (\(H_{S-O}\propto Z^2\))
  - - - E=-R_y*(Z/n)^2 not§ l,m
        => aber rel corr. => §l, ext field => §m
      - R_n.(0)=0 for l!=0
      - number of nodes in R_nl: n-l-1
      - only solutions with n>l
  - - - ground state:
        
        neglect intterm (ind zi model): psi=psi(r1)+psi(r2)
        
        perturbation; <psi_1s2|1/r_12|psi_1s2> ∆E=5/4•R_y :red_cross:
      - Hartree-Fock-method :red_cross:
      - excited state:
        1.neglect intterm ->E=-Z^2•R_y (1/n1^2+1/n2^2)
        
        one close to nucleus(Z'=Z) one far (Z'=Z-1)
        
        exchange op: P_12 commutes with H
        => ∆E=direct±exchange integral: direct:repulsion, exchange=correlation: psiA(r1=r2=0->∆E lower
        => sym is ES
        no two-fold excited helium =>above ionization threshold
      - spin: P_12->sym(triplett),antisym(singulett)
        Pauli-Prinzip
        
        optical transitions: ∆S=0 (dipole operator acts not on spin space)
        
        substructure: spin-orbit-int (J=L+S)
- - - - 1."rigid" molecule
        neglect kin E of protons (\( M>>m\))
        fix R , minimize E(R)
      - 2.reintroduce T of protons
        solve SE for nuclear motion
    - - \(\psi_{r,R}=c_1\psi_A+c_2\psi_B\)
        symmetry requires \(\psi_{g,u}=\frac{1}{\sqrt{2\pm2S_{AB}}}(\psi_A\pm\psi_B)\)
        \(\psi_{A,B}\) wavefunc of e in ground state of atom A, B