Post on 31-Dec-2015
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