Physics 451

Quantum mechanics I

Fall 2012

Dec 5, 2012

Karine Chesnel

Homework

Quantum mechanics

Last assignment

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

Final exam

Wednesday Dec 12, 20127am – 10am

C 285

Class evaluation

Please fillthe class evaluation

survey online

Quiz 34: 5 points

Quantum mechanics

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

Free electron gas

Quantum mechanics

Bravaisk-spacexk

yk

zk

Fk Fermi 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

Fk Fermi surface

Solid Quantum pressure

2

3tot tot

dVdE E

V

2/32 2

5/332

3 5totE

PV m

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 k x a B k x a or

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

Quiz 33Quantum mechanics

A. Conductor

B. Insulator

C. Semi-conductor

A material has q=3 valence electrons / atoms.In which category will it fall

according to the 1D dirac periodic potential model?

Final ReviewQuantum mechanics

What to remember?

Quantum mechanics

Wave function and expectation values

*x x dx

“Operator” x

*p i dxx

“Operator” p

* ,Q Q x i dxx

Quantum mechanics

Time-independent Schrödinger equation

2 2

22i V

t m x

Here ( )V xThe potential is independent of time

/( , ) ( ) ( ) ( ) iEtx t x t x e Stationary state

/

1

( , ) ( ) niE tn n

n

x t c x e

General state

Review IQuantum mechanics

Infinite square well

2 2 2

22n

nE

ma

Quantization of the energy

2sinn

nx

a a

x0 a

Ground state 1 1,E

Excited states

2 2,E

3 3,E

Quantum mechanics

Harmonic oscillator

x

V(x)

1

2a ip m x

m

1 1

2 2H a a a a

2 2 21 1( )

2 2V x kx m x

• Operator position 2

x a am

• Operator momentum 2

mp i a a

Review IQuantum mechanics

4. Harmonic oscillator

Ladder operators:

0

1

!

n

n an

nnE

Raising operator: 11n na n nE

a1n

Lowering operator: 1n na n nE

a1n

1

2nE n

Quiz 35Quantum mechanics

A.

B.

C.

D.

E. 0

What is the result of the operation ? 4 3a

77

23

04!

03!

Quantum mechanics

Square wells and delta potentials

V(x)

x

Bound statesE < 0

ScatteringStates E > 0

Symmetry considerations

even evenx x

odd oddx x

Physical considerations

ikxreflected x Be

ikxincident x Ae

ikxtransmitted x Fe

Quantum mechanics

Square wells and delta potentials

Ch 2.6

Continuity at boundaries

Delta functions

Square well, steps, cliffs…

dx

d

is continuous

is continuous except where V is infinite

022

m

dx

d

dx

d

is continuous

is continuous

Quantum mechanics

Scattering state 0E

0

ikx ikxleft x Ae Be ikx

right x Fe

A F

Bx

Reflection coefficient Transmission coefficient

2 2

1

1 2 /R

E m 2 2

1

1 / 2T

m E

The delta function well/ barrier

V x x

“Tunneling”

Formalism

Quantum mechanics

ˆijH H Linear transformation

(matrix)Operators

Wave function Vector

Observables are Hermitian operators †Q Q

Q̂ a a a is an eigenvector of Q

is an eigenvalue of Q

Quantum mechanics

Eigenvectors & eigenvalues

0T I a

det 0T I

To find the eigenvalues:

We get a Nth polynomial in : characteristic equation

Find the N roots 1 2, ,... N Spectrum

Find the eigenvectors 1 2, ,... Ne e e

Quantum mechanics

The uncertainty principle

,2A B

A B

i

Finding a relationship between standard deviations for a pair of observables

Uncertainty applies only for incompatible observables

Position - momentum 2x p

Quantum mechanics

The uncertainty principle

Energy - time

2E t

Special meaning of t

Qtd Q

dt

,d Q i QH Q

dt t

Derived from the Heisenberg’s equation

of motion

Quiz 33Quantum mechanics

A.

B.

C.

D.

E.

Which one of these commutation relationships is not correct?

,x p i

,y z xL L i L

2, 0xL L

( ),V x x o

,H x o

Quantum mechanicsSchrödinger equation in

spherical 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

r2

2 ( )2

H V r Em

Quantum mechanics

The hydrogen atom

11( ) ( )lR r e v

r

Quantization of the energy

22 2

2 2 2 20

1 1

2 4 2n

m eE

n ma 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

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

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

Periodic table

Quantum mechanics

Filling the shells

1 2 2 3 3 4 ...s s p s p s2 2 6

Periodic table

Quantum mechanics

1 2 2 3 3 4 ...s s p s p s

2 1SJL

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

yk

zk

Bravaisk-space

xk

xk

yk

zk

FkFermi surface

e-

•Free electron gas theory

• Crystal Bloch’s theory

2 2 2

2/3232 2

FF

kE

m m

Quantum mechanics

Thank you for your participation!

And Merry Christmas!

Good luck for finals