Physics 451

Quantum mechanics I

Fall 2012

Nov 20, 2012

Karine Chesnel

EXAM III

Quantum mechanics

• Time limited: 3 hours• Closed book• Closed notes• Useful formulae provided

When: Monday Nov 26 – Fri Nov 30Where: testing center

EXAM III

Quantum mechanics

1. Schrödinger equation in spherical coordinates

2. Hydrogen atom: spherical harmonics

3. Hydrogen atom: radial function and energy

4. Electron’s spin and Pauli matrices

5. Spin and Magnetic field, Combination of two spins

Quantum mechanics

Schrödinger equation in spherical coordinates

22 ( )

2H V r E

m

22 2

2 2 2

1 1 1sin

sin sinr

r r r

Quantum mechanics

Schrödinger equation inspherical coordinates

2

2 2

1 1 1sin ( 1)

sin sin

Y Yl l

Y

The angular equation

, , ,mnlm nl lr R r Y

The radial equation 2

22

1 2( ) ( 1)

d dR mrr V r E l l

R dr dr

x

y

z

r

Quantum mechanics

The hydrogen atom

11( ) ( )lR r e v

r

Quantization of the energy22

2 20

1

2 4n

m eE

n

max

0

( )j

jj

j

v c

1

2( 1 )

( 1)( 2 2)j j

j l nc c

j j l

Bohr radius2

1002

40.529 10a m

me

kr

Quantum mechanics

The hydrogen atom

21

n

EEn Energies levels

Spectroscopy

221

11

fi nnE

hcE

Energy transition

22

111

if nnR

Rydberg constant

E0

E1

E2

E3

E4

Lyman

Balmer

Paschen

Quantum mechanics

x

y

z

r

Normalization

, , ,r R r Y

2

r dr

2 sindr r drd d

2 2

0

1r

R r r dr

Radial part

22

0 0

( , ) sin 1d Y d

angular part

Quantum mechanics

The angular momentumeigenvectors

x

y

z

r

Spherical harmonicsare the

eigenfunctions

nlm n nlmH E

2 2 ( 1)nlm nlmL l l

z nlm nlmL m

2 22

1

2r L V E

mr r r

Quantum mechanics

The spin

2 2 ( 1)S sm s s sm

zS sm m sm

( 1) ( 1) 1S sm s s m m s m

Quantum mechanics

Pauli matrices

2 2 1 03

0 14S

0 1

1 02xS

0

02y

iS

i

1 0

0 12zS

x y z

Quantum mechanics

Adding spins S

Possible values for S when adding spins S1 and S2:

1 2 1 2 1 2 1 2, 1 , 2 ,...S S S S S S S S S

1 2

1 2

1 2

1 1 2 2s s sm m m

m m m

sm C s m s m

Clebsch- Gordan coefficients

EXAM II

2 2

22i V

t m x

Quantum mechanics

Some of the formulae provided

Schrödinger equation2

2 22 2 2

1 1 1sin

sin sinr

r r r

2

04

eV

r

202

4a

me

Potential in hydrogen atom: and Bohr radius:

22 *

' ' ' ' ' '

0 0 0

sin nlm n l m nn ll mmd d drr

2

* '' ' '

0 0

sin l lm m ll mmd d Y Y

22

0

( ) 1drr R r

Normalization

( , )mllm Y zL lm m lm 2 2 ( 1)L lm l l lm

x yL L iL ( 1) ( 1) 1L lm l l m m l m

Angular momentum

Quantum mechanics