physics 452 quantum mechanics ii winter 2012 karine chesnel
TRANSCRIPT
Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel
Homework
Phys 452
Wed Feb 16: assignment # 10Pb 7.8, 7.9, 7.13, 7.17
Friday Feb 18: assignment # 118.1, 8.2, 8.7, 8.14
Announcements
Monday: Holiday
Next class: Tuesday at 1pm
H atom
Hydrogen molecule ion H2+
Phys 452
H atom
electron
1r
2r
R
Hydrogen molecule ion H2+
Phys 452
electron
1r
2r
R
2 2
0 1 2
1 1
2 4
p eH
m r r
Step 1: Hamiltonian
Step 2: trial wave function 0 1 0 2A r r
Overlap integral: 0 1 0 2r r
Normalization: 1/2
2/1 1
1 132
R a R RA e
a a
0 1r
R
0 2r
Hydrogen molecule ion H2+
Phys 452
0 1 0 2A r r
Use the trial wave function:
2 2
0 1 0 20 1 2
1 1
2 4
p eH A r r
m r r
2 22 2 2 2
0 1 0 2 0 1 0 20 1 0 2 0 2 0 1
1 1 1 1
2 4 2 4 4 4
p pe e e eH A r r r r
m r m r r r
Hydrogen Hamiltonianwith first nucleus
Hydrogen Hamiltonianwith second nucleus
Cross terms
Step 3: expectation value of H
0 1r
R
0 2r
Hydrogen molecule ion H2+Phys 452
2
1 0 1 0 20 2 1
1 1
4n
eH E A r r
r r
2
2
1 0 1 0 1 0 2 0 2 0 1 0 2 0 2 0 10 2 1 1 2
1 1 1 1
4n
eH H E A r r r r r r r r
r r r r
2
2
1 0 1 0 1 0 1 0 20 2 1
1 12
4n
eH H E A r r r r
r r
Direct integral D exchange integral X
Pb. 7.8
Eigenstates of Individual hydrogen atoms
00 1 0nH E
0 1r
R
0 2r
Hydrogen molecule ion H2+
Phys 452
11 2
1
D XH E
I
where
2/ 1
13
R a R RI e
a a
2 /1 R aa aD e
R R
/1 R aaX e
R
directintegral
exchangeintegral
Finally…!
0 1r
R
0 2r Hydrogen molecule ion H2
+Phys 452
1 1
21 2
1
D X aH E E
I R
Step 4: Minimization
First include the proton-proton interaction !
2
0
1
4pp
eV
R
2 2
21
1 (2 / 3) 121
1 1 1/ 3
x x
x
x e x eHF x
E x x x e
wherex = R/a
0 1r
R
0 2r Hydrogen molecule ion H2
+Phys 452
Step 4: Minimization
Rx
a
1
H
E
Presence of a minimum:Evidence of bonding
Equilibrium separation distance:
2.4 1.3eqR a
Quiz 15
Phys 452
The binding energy for the hydrogen molecule ion H2+
is experimentally found to be 2.8eV.What can we predict about the binding energy E
estimated with the variational principle?
A. E > 2.8 eV
B. E < 2.8 eV
C. E = 2.8 eV
D. Can be any value
E. Can not tell
0 1r
R
0 2r
Hydrogen molecule ion H2+
Phys 452
Pb 7.8
Calculation of
Direct integral 0 1 0 12
1D r r
r
0 1 0 21
1X r r
r
Exchange integral
0 1r
R
0 2r Hydrogen molecule ion H2
+Phys 452
Pb 7.9
Rx
a
1
H
E
Presence of a minimum:Evidence of bonding
For symmetrical state
0 1 0 2A r r
What about antisymmetrical state?
0 1 0 2A r r
0 r Hydrogen atom H
Phys 452
Pb 7.13
Another trial wave function
2brr Ae
Hamiltonian 2
2
pH V r
m
Calculate H …and minimize it
use spherical coordinates
0 2r Helium - like systemPhys 452
Pb 7.17
“Rubber – band” model for He
0 1r
2 2
22 2 2 21 21 2 1 2
1
2 2 2 4
p pH m r r m r r
m m
a) Change of variable
b) Exact solution (harmonic oscillators)
c) Evaluate with ground state of 3D HO 0 0H 0