Physics 451

Quantum mechanics I

Fall 2012

Dec 3, 2012

Karine Chesnel

Homework

Quantum mechanics

Last two assignment

• HW 23 Tuesday Dec 45.9, 5.12, 5.13, 5.14

• HW 24 Thursday Dec 65.15, 5.16, 5.18, 5.19. 5.21

Wednesday Dec 5Last class / review

Periodic table

Quantum mechanics

Hund’s rules2 1S

JL

• First rule: seek the state with highest possible spin S(lowest energy)

• Second rule: for given spin S, the state with highest possible angular momentum L has lowest energy

• Third rule: If shell no more than half filled, the state with J=L-S

has lowest energy If shell more than half filled, the state with J=L+S

has lowest energy

Quiz 32aQuantum mechanics

What is the spectroscopic symbol for Silicon ?

A.

B.

C.

D.

E.

Si: (Ne)(3s)2(3p)2

21S

32P

30P

42S

42D

Quiz 32bQuantum mechanics

What is the spectroscopic symbol for Chlorine ?

A.

B.

C.

D.

E.

Cl: (Ne)(3s)2(3p)5

21S

23/2P

30P

42S

42D

SolidsQuantum mechanics

e-

What is the wave function

of a valence electron in the solid?

Solids

Quantum mechanics

e-Basic Models:

• Free electron gas theory

• Crystal Bloch’s theory

Free electron gas Quantum mechanics

e-

e-

lz

ly

lx

Volume x y zV l l l

Number of electrons: Nq

Free electron gas

22 ( )

2H V r

m

Quantum mechanics

( , )r t

e-

3D infinitesquare well

, ,V x y z 0 inside the cube

outside

22

2E

m

Free electron gas

Quantum mechanics

e-

22

2E

m

Separation of variables

( , ) ( ) ( ) ( )x y zr t x y z 2

2

2 i i i iEm

8( , ) sin sin sinyx z

x y z x y z

n yn x n zr t

l l l l l l

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

Free electron gas

Quantum mechanics

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

xk

yk

zk

Bravaisk-space

Free electron gas

Quantum mechanics

xk

yk

zk

Bravaisk-space

Fk Fermi surface

Free electron densityNq

V

1/323Fk

Free electron gas

Quantum mechanics

Bravaisk-spacexk

yk

zk

FkFermi surface

2 2 2

2/3232 2

FF

kE

m m

Total energy contained inside the Fermi surface

2 52/3

20 0 10

F FE k

Ftot k k

k VE dE E n dk V

m

Free electron gas

Quantum mechanics

Bravaisk-spacexk

yk

zk

FkFermi surface

Solid Quantum pressure

2

3tot tot

dVdE E

V

2/32 2

5/332

3 5totE

PV m

Solids

Quantum mechanics

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

e-

yk

zk

Bravaisk-space

xk

xk

yk

zk

FkFermi surface

Number of unit cells 236.02 10AN

SolidsQuantum mechanics

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

e-

yk

zk

Bravaisk-space

xk

xk

yk

zk

FkFermi surface

Pb 5.15: Relation between Etot and EF

Pb 5.16: Case of Cu: calculate EF , vF, TF, and PF

SolidsQuantum mechanics

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

e-

yk

zk

Bravaisk-space

xk

xk

yk

zk

FkFermi surface

Number of unit cells 236.02 10AN

Solids

Quantum mechanics

V(x)

( ) ( )V x a V x

Dirac comb

Bloch’s theorem

( ) ( )iKax a e x 2 2

( ) ( )x a x

1

0

( ) ( )N

j

V x x ja

Solids

Quantum mechanics

V(x)

( ) ( )x Na x

Circular periodic condition

1iNKae

2 nK

Na

x-axis “wrapped around”

Solids

Quantum mechanics

V(x)

( ) sin( ) cos( )x A kx B kx

Solving Schrödinger equation

0 a

2 2

22

dE

m dx

0 x a

Solids

Quantum mechanics

V(x)

( ) sin( ) cos( )x A kx B kx

Boundary conditions

0 a

0 x a

( ) ( )iKax a e x 0a x

( ) sin( ) cos( )iKax e A kx B kx

Solids

Quantum mechanics

V(x)

( ) sin( ) cos( )right x A kx B kx

Boundary conditions at x = 0

0 a

• Continuity of

• Discontinuity of d

dx

sin( ) cos( )iKae A ka B ka B

2

2cos( ) sin( )iKa m

kA e k A ka B ka B

( ) sin ( ) cos ( )iKaleft x e A k x a B k x a

Solids

Quantum mechanics

2cos( ) cos( ) sin( )

mKa ka ka

k

Quantization of k:

sin( )( ) cos( ) cos( )

zf z z Ka

z

z ka

2

m a

Band structure

Pb 5.18Pb 5.19Pb 5.21

Quiz 33Quantum mechanics

A. 1

B. 2

C. q

D. Nq

E. 2N

In the 1D Dirac comb modelhow many electrons can be contained in each band?

Solids

Quantum mechanics

Quantization of k: Band structure

E

N states

N states

N states

Band

Gap

Gap

Band

Band

(2e in each state)

2N electrons

Conductor: bandpartially filled

Semi-conductor: doped insulator

Insulator: bandentirely filled

( even integer)q