The hydrogen

Consists of two Physical particles, a positively charged nucleus and a negatively charged Lecture 16 electron Hamiltonian has three The Hydrogen Atom, a Central- terms  Nuclear kinetic Problem H  Tn  Te  Vne  Electronic kinetic energy p2 p2 1 e2  Nuclear-electron  n  e  Coulombic potential 2mn 2me 40 | rne | energy

Simplification of the hydrogen- Center of mass atom problem

By substitution, the hydrogen-atom Hamiltonian as a Hamiltonian becomes function of nuclear 2 1 2 1 2 1 e and electronic co- H  PCM  p  2M 2 4 r ordinates is complex 0  H  H m m CM rel R  1 r  2 r Define CM M 1 M 2

 Center of mass, RCM r  r2  r1 Decompose into two problems

 Relative position, r M  m1  m2 m m HCM CM  ECM CM H rel rel  Erel rel  Total mass, M   1 2 m1  m2  Reduced mass,  p2 p2 P2 p2 1  2  CM    CM rel E  ECM  Erel 2m1 2m2 2M 2

Center-of-mass problem The relative problem

A particle of mass, M, moving in no Consider relative H  E of nucleus and electron, 2 1 e2 potential   2    E with only Coulombic 2 40 r Like the particle-in-a-box problem interaction

Energies and wave functions known 2 The dependence of  2 1     1 1 e2   r 2   L2   from that model 2 2 on angular variables 2 r r  r  2r 40 r Translational degrees of freedom of the simplifies the problem  E 2  Eigenfunctions of L are (r,,)  R(r)Y ( ,) hydrogen atom well known m

1 Hydrogenic and The radial equation eigenfunctions Use of the spherical harmonic functions gives an Products of spherical harmonic functions with equation for the radial part functions related to Laguerre polynomials 2 1   R  1 2 R 1 e2    r 2       R  ER  (r, ,)  A r / a  L (r / a )exp(r / na )Y ( ,) 2 2 nm nm  0  n 0 0 m 2 r  r r  2 r 40 r This can be put into dimensionless form by defining Laguerre polynomials n L (x) the radial distance in bohrs  n 2 1 0 1  h 2 0 2 - x 0 2 1 1 r  a0 a0  2 3 0 3 – 2x + 2x2/9 mee 3 1 4 – 2x/3 The dimensionless equation is related to Laguerre’s 3 2 1 differential equation Energies of the state depend only on n 2  1 e 1 2 Rn ( )  A Ln ( )exp / n   En   2 gn  n 40 2a0 n

Energies Hydrogen-atom energy levels Defined relative to Can give energies in the “vacuum level” “macroscopic units”  Energy level in  Joules which the electron and the nucleus are  Ergs just pulled apart Often convenient to States are labeled use “atomic units” by n and 1 hartree  27.2114 electron volts   Hartree  States of different 1 rydberg  13.606 electron volts   Rydberg given by letters  0.5 hartree  s ( = 0)  Electron volt 1electron volt  96.4853 kilojoules / mole  p ( = 1)  d ( = 2)

Addition of spin Electron spin

The spatial part is incomplete S = ½ One incorporates spin as a separate co- Two states ordinate  m = ½ or  Wave function is a  m = - ½ or  Eigenstates of the spin  (r, ,)  A R (r / a )Y ( ,) angular momentum n,l,m,I ,mI nl 0 lm spin,S ,ms operators 2 2 Energy is unchanged by addition of spin  S  = ½(½+1)  2 2 variables  S  = ½(½+1)   S = ( /2) Increases degeneracy z     Sz  = - (/2) 

2 Summary Central-force (hydrogen-atom) problem separates  Center of mass movement (translation)  Relative motion  Angular part is constant-angular-momentum problem  Radial part is related to Laguerre’s equation Energy depends on principal quantum number, n, as predicted by Rydberg Degeneracy  States with same n but different angular momentum quantum numbers,  and m, have same energy 2  gn = n

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