Physics 451

Quantum mechanics I

Fall 2012

Nov 12, 2012

Karine Chesnel

Announcements

Quantum mechanics

Homework this week:

• HW #19 Tuesday Nov 13 Pb 4.14, 4.15, 4.16, 4.17

• HW #20 Thursday Nov 15Pb 4.18, 4.19, 4.21, 4.22

Test 3 Review Monday Nov 19- 20Sign for practice test

Quantum mechanics

The angular momentum

x yL L iL Ladder operator

L

L

2 2z zL L L L L

TopValue=+l

BottomValue = -l

Eigenstates m ml lf Y

2 2 ( 1)m ml lL f l l f

m mz l lL f mf

Quantum mechanics

The angular momentum

L

L

2 2 ( 1)m ml lL Y l l Y

m mz l lL Y mY

1m m ml l lL Y Y

Pb 4.18

1( 1) ( 1)m ml lL Y l l m m Y

Quantum mechanics

The angular momentumIn spherical coordinates

1

sinL r r r r r

i r

x

y

z

r

1

sinL

i

zL i

L r p ri

Quantum mechanics

The angular momentumIn spherical coordinates

x

y

z

r

cotiL e i

x yL L iL

22 2

2 2

1 1sin

sin sinL

Pb 4.21, 4.22

Quantum mechanics

The angular momentumeigenvectors

x

y

z

r

m m mz l l lL Y Y m Y

i

22 2 2

2 2

1 1sin ( 1)

sin sinm m ml l lL Y Y l l Y

and

were the two angular equations for the spherical harmonics!

Spherical harmonicsare the

eigenfunctions

nml n nmlH E

2 2 ( 1)nml nmlL l l

z nml nmlL m

Quantum mechanics

The angular momentumand Schrödinger equation

x

y

z

r

2 22

1

2r L V E

mr r r

3 quantum numbers (n,l,m)

• Principal quantum number n: integer• Azimutal and magnetic quantum numbers (l,m)

can also be half-integers.

Quantum mechanics

Quiz 26

A. 0

B.

C.

D.

E.

For a given n value, how many eigenstates can we find for

the operator ?

2 1n

2n

n

( 1)n n

2L

Quantum mechanics

The Spin

Types of angular momentum

L r p orbital

L I spin

Quantum mechanicsAgular moment in the atom

• Orbital moment (l)

Representation of ,,rnlm

• Spin moment (s)

Quantum mechanics

Spin in elementary particles

Each elementary particle is characterized by an immutable spin S

• Fermions: (S half-integer)

• Bosons: (S integer)

S=1/2Leptons: electrons,…

Quarks: u,b,c,s,t,b

Proton, neutron

Mesons

Photon S=1

Quantum mechanics

The spin

ˆL̂ S

,

,

,

x y z

y z x

z x y

S S i S

S S i S

S S i S

2 2 2, , , 0x y zS S S S S S

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

The spin 1/2

2 23

4S

2zS

The “spinor”

2 23

4S

2zS

2 2 1 03

0 14S

1 0

0 12zS

1

0

Spin up

0

1

Spin down

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

Pb 4.29