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