Atomic Physics

Atomic Physics

1. Lorentz model

   • Electron is like a damped, driven (by light) harmonic oscillator.
   • Absorption profile: average absorption from E field
     • Write in terms of the Lorentzian by performing approximation:w^2 - w_0^2 = 2*w*(w - w_0)
   • Classical lifetime (goes as 1/w^2) => wrong freq. dependence

2. Einstein coefficients

   • Consider 2 state-system.
   • Rate of change of number of particles in excited state consists of
     • Light driven components: P(transition) proportional to density of radiation
       • Emission: B_21
       • Absorption: B_12
     • Spontaneous decay: rate = A_12
     • At equilibrium it must be 0
       • Equating rho to Planck's expression for BB radiation fixes:
         • B_21 = B_12
         • relation between A_21 and B_21
         • B_21 is a property of the atom (see LN7)
     • Selection rules
       • Tell which transitions are allowed.
       • l = l' +-1
       • m = m' along z m = m' +-1 along x,y
       • s = s'

3. Hydrogen atom

   • Classical 2-body problem
     • Consider centre of mass frame, r = relative displacement and u= reduced mass
       • Calculate potential and kinetic energy
   • Quantum problem
     • Start with classical Hamiltonian in spherical polars
       • Treat L^2 as a parameter as it is conserved
         • Separable solution: radial eq'n (use e.vals of L^2: l(l+1)h^2)
           • Substitute variables and constants
             • Exponential behaviour at large p
             • Power behaviour at small p
             • Multiply by series sol'n: find recursion rel'n
               • Terminate series for quantisation
     • Sol'n has 3 quantum n's: nlm
       • l=0 have no ang.mom: no angular dependence
       • m goes from -l to +l
       • l goes from 0 to n
       • Spectroscopic notation
         • Letters represent l values
           • s = 0, p = 1, d = 2, f = 3, g, h, ...

4. 2-particle states

   • Helium atom
     • Add two Hamiltonians, ignore e-e repulsion term
       • Wavefunction is antisymmetrised product of 2 states
   • Wavefunction of a 2-p. system is product of single wavefunctions
     • Can simply attach the spin part to the wavefunction
       • For fermions, spin statistics theorem demands an overall antisymmetric wavefunction