Download Physical Chemistry The hydrogen atom Center of mass

The hydrogen atom
Consists of two
particles, a positively
charged nucleus and a
negatively charged
electron
Hamiltonian has three
terms
Physical Chemistry
Lecture 16
The Hydrogen Atom, a CentralForce Problem
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Center of mass, RCM
Relative position, r
Total mass, M
Reduced mass, 
H
 Tn

 Te
pn2
2mn

 Vne
pe2
2me

1 e2
4 0 | rne |
Simplification of the hydrogenatom problem
Center of mass
Hamiltonian as a
function of nuclear
and electronic coordinates is complex
Define
Nuclear kinetic energy
Electronic kinetic energy
Nuclear-electron
Coulombic potential
energy
By substitution, the hydrogen-atom
Hamiltonian becomes
H
m1
r1 
M
 r1

R CM
r  r2
M
 m1
p12
2m1


H CM CM
2
PCM
2M
1 2
p
2


1 e2
4 0 r
 H rel
Decompose into two problems
 m2
p22
2m2
1 2
PCM
2M
 H CM
m2
r2
M
m1m2
m1  m2
 


H rel rel
  CM rel
p2
2
Center-of-mass problem
A particle of mass, M, moving in no
potential
Like the particle-in-a-box problem
Energies and wave functions known
from that model
Translational degrees of freedom of the
hydrogen atom
 ECM CM
 Erel rel
E  ECM

Erel
The relative problem
Consider relative motion
of nucleus and electron,
with only Coulombic
interaction
The dependence of 2
on angular variables
simplifies the problem

Eigenfunctions of L2 are
well known
H  E


2 2
  
2
1 e2
  E
4 0 r
1 2
 2 1   2  
L 
r
 
2  r 2 r  r 
2 r 2
 E
1 e2

4 0 r
 (r , ,  )  R (r )Ym ( ,  )
1
Hydrogenic energies and
eigenfunctions
The radial equation
Use of the spherical harmonic functions gives an
equation for the radial part

 1   2 R 
 
 r
2 r 2  r r 
2
  1
2
2
R
r2

2
1 e
R  ER
4 0 r
This can be put into dimensionless form by defining
the radial distance in bohrs
r
  a0
a0


En
 

1 e2 1
4 0 2a0 n 2
gn
 n2
1 hartree  27.2114 electron volts
1 rydberg  13.606 electron volts
Energy level in
which the electron
and the nucleus are
just pulled apart
States are labeled
by n and 

 0.5 hartree
States of different 
given by letters
 s ( = 0)
 p ( = 1)
 d ( = 2)
1 electron volt  96.4853 kilojoules / mole
Addition of spin
The spatial part is incomplete
One incorporates spin as a separate coordinate
Wave function is a product
n ,l ,m , I ,mI (r , ,  ) 
Ln(x)
1
2-x
1
3 – 2x + 2x2/9
4 – 2x/3
1
Energies of the state depend only on n
Joules
Ergs
Hartree
Rydberg
Electron volt

0
0
1
0
1
2
Defined relative to
the “vacuum level”
Often convenient to
use “atomic units”
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n
1
2
2
3
3
3
Hydrogen-atom energy levels
Can give energies in
“macroscopic units”
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
Laguerre polynomials
A  Ln ( ) exp  / n 
Energies

Anm r / a0  Ln ( r / a0 ) exp(r / na0 )Ym ( ,  )
nm (r , ,  ) 
 0h2

me e 2
The dimensionless equation is related to Laguerre’s
differential equation
Rn ( ) 
Products of spherical harmonic functions with
functions related to Laguerre polynomials
A Rnl (r / a0 )Ylm ( ,  ) spin , S ,ms
Energy is unchanged by addition of spin
variables
Increases degeneracy
Electron spin
S=½
Two states
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m = ½ or 
m = - ½ or 
Eigenstates of the spin
angular momentum
operators
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S2  = ½(½+1)2 
S2  = ½(½+1)2 
Sz  = (/2) 
Sz  = - (/2) 
2
Summary
Central-force (hydrogen-atom) problem
separates
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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
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States with same n but different angular
momentum quantum numbers,  and m, have
same energy
gn = n2
3