Physical Chemistry the Hydrogen Atom Center of Mass Simplification

The hydrogen atom Consists of two Physical Chemistry particles, a positively charged nucleus and a negatively charged Lecture 16 electron Hamiltonian has three The Hydrogen Atom, a Central- terms Nuclear kinetic energy Force Problem H Tn Te Vne Electronic kinetic energy p2 p2 1 e2 Nuclear-electron n e Coulombic potential 2mn 2me 40 | 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 motion H E of nucleus and electron, 2 1 e2 potential 2 E with only Coulombic 2 40 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 2r 40 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 energies 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 nm nm 0 n 0 0 m 2 r r r 2 r 40 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 40 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 product 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 3.

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