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An Electron Trapped in A Potential Well
Probability densities for an infinite well
2/2/,0)( LxLxU
2/,2/,)( LxLxxU
)()()(d
)(d
2 2
22
xExxUx
x
m
Solve Schrödinger equation
2/ ,2/)()(d
)(d
2 2
22
LxLxxExx
x
m
outside the well
2/ ,2/0)( LxLxx
)(xU
x2/Lo2/L
2 2
2
d ( )( ) / 2 / 2
2 d
xE x L x L
m x
inside the well
22
2
d ( )( ) / 2 / 2
d
xk x L x L
x
2
2
mE
k
An Electron Trapped in A Potential Well
kxBkxAx cossin)( The general solution is
)(xU
x2/Lo2/L
By the boundary conditions 0)2/cos()2/sin( kLBkLA 0)2/cos()2/sin( kLBkLA
0)2/cos()2/sin( kLBkLA
0)2/cos(2 kLB
0)2/sin(2 kLA nkL 0 ,6,4,2 Bn x)
L
nπ(Asin
x)L
nπ(Bcos 0 ,5,3,1 An
0)2/cos(or
0)2/sin(
kL
kL
An Electron Trapped in A Potential Well)(xU
x2/Lo2/L
0 ,6,4,2 Bn x)L
nπ(Asin
x)L
nπ(Bcos 0 ,5,3,1 An
Normalization
2/
2/1d)(*)(
L
Lxxx
For odd wave function )cos()( xL
nBxodd
/2 2 2
/2cos ( ) 1
L
L
nB x dx
L
L
B2
For even wave function
/2 2 2
/2sin ( ) 1
L
L
nA x dx
L
2A
L
2sin( )
nx
L L
2cos( )
nx
L L
1
2( ) cos
xx
L L
3
2 3( ) cos
xx
L L
2
2 2( ) sin
xx
L L
probability
2 2
2
d ( )( ) / 2 / 2
2 d
xE x L x L
m x
inside the well
22
2
d ( )( ) / 2 / 2
d
xk x L x L
x
2
2
mE
k
An Electron Trapped in A Potential Well)(xU
x2/Lo2/L
0)2/cos(2 kLB
0)2/sin(2 kLA nkL
L
nk
2
22222
22 mL
n
m
kE
1
2( ) cos
xx
L L
3
2 3( ) cos
xx
L L
2
2 2( ) sin
xx
L L
2
22
1 2mLE
2
22
2
2
mLE
2
22
3 2
9
mLE
energy distribution )(xU
x2/Lo2/L
0U
Barrier TunnelingBarrier Tunneling
Dividing the space into three parts: I) to the left of the barrier II) within the barrier; and III) to the right of the barrier
)0( )( 21 xeex ikxikxI
)( )( 5 Lxex ikxIII
22
02
d ( )( ) ( )
2 dII
II II
xU x E x
m x
2 0
2 2
2 ( )d ( )( ) 0
dII
II
m U Exx
x
20 /)(2 EUmII
0)(d
)(d 22
2
xkx
xII
II
xixiII
IIII eex 43)(IIi
xxII eex 43)(
The conditions for wave functions at the boundary are continuity.
Lx
III
Lx
II
IIIII
x
II
x
I
III
x
x
x
x
LL
x
x
x
x
d
)(d
d
)(d
)()(
d
)(d
d
)(d
)0()0(
00
LeU
E
U
ET 2
00
)1(16
Barrier TunnelingBarrier Tunneling
LeU
E
U
ET 2
00
)1(16
Tunneling current
Si(111) Surface
mm 11
Tetracene/Ag(110)
[001]
0.00
0.05
0.10
Hig
ht(
nm
)
0.04 nm
Wave (matter wave)
Uncertainty Principle
Wave Function
SchrSchrödinger’södinger’s EquationEquation
probability
Free ElectronsHydrogen atomHydrogen atom
p
h
Ev
h de Broglie relation,
de Broglie wavelength.
/ 2xx p / 2E t
)(0),( tkxieψtxΨ
),(*),(),(2
txψtxψtxΨ
Potential Well
Particle
The Nature of The Nature of MatterMatter
Energy quantization Probability
Barrier TunnelingBarrier Tunneling STM
Chapter 48
Atomic Structure
r
SchrSchrödinger’södinger’s EquationEquation
)()()()())(1
(2 2
22
2
2
rErrUrr
l
rr
rrm
r
zeU
2
04
1
2
0
1( ) ( )
4
zer r
r
0l
0/
30
1)( are
ar
2
20
0 me
ha
=0.529ÅBohr radius
0/230
2 1)( are
ar
drre
adVr ar 2/2
30
2 41
)( 0
drrp )(
0)41
( 2/230
0 readr
d ar
0ar
ground state (n=1)n=2
1l
4
2 2 20
1 1, 2, 3,
8n
meE n
h n
11, 13.6eVn E
12n
EE
n
ground state
Energy quantization
Principle quantum numberr
zeU
2
04
1
r
SchrSchrödinger’södinger’s EquationEquation
)()()()())(1
(2 2
22
2
2
rErrUrr
l
rr
rrm
2
0
1( ) ( )
4
zer r
r
The Uncertainty Principle
2xp x
2L
rpL
rΦx
The angular momentum-angle Uncertainty Relationship
L p r
r
SchrSchrödinger’södinger’s EquationEquation
Angular Momentum of Electrons in Atoms
)1( llL
)1( ,3 ,2 ,1 ,0 nlAngular momentum quantum number
)()()()())(1
(2 2
22
2
2
rErrUrr
l
rr
rrm
Angular momentum quantization
l 0 1 2 3 4 5
code s p d f g h
Using this labeling, we can express 1s for ground state (n=1, l=0). The first excited state has two designations: 2s (n=2, l=0) and 2p (n=2, l=1).
Angular momentum quantum number
l=0, 1, 2, …, n-1. Ex. n=1, l=0 for s state
n=2, l=0, 1. for l=0, s state and l=1, p state
Space quantization
)1(coscos 11
ll
m
L
L lz
2)1( ,1 llLl
θml=-l, -(l-1), …-1, 0, 1, …,(l-1), l
magnetic quantum number
0)1( ,0 llLl
z
,3 ,2 ,1 1
8 2220
4
nnh
meEn
n Principle quantum number
)1( llL )1( ,3 ,2 ,1 ,0 nl
l Angular momentum quantum number
ml=-l, -(l-1), …-1, 0, 1, …,(l-1), l
ml magnetic quantum number
)1(coscos 11
ll
m
L
L lz
,....,,, NMLK
,....,,, fdps
(n=1, l=0)The Ground State
ground state (n=1)
ml=0, is spherically symmetric.
drrea
dVr ar 2/230
2 41
)( 0
0)41
( 2/230
0 readr
d ar
0ar
The 2s State (n=2, l=0)
ml=0, is spherically symmetric.
An Excited State Of The Hydrogen Atom
The 2p State (n=2, l=1)
ml=-1, 0, +1 is not spherically symmetric.
Electron Spin
Pauli pointed out the need for a 4th quantum number in 1924—Spin quantum number
2
1sand)1( ssS
)1( llL
l Angular momentum quantum number
Spin magnetic quantum number2
1sm
)1( ,3 ,2 ,1 ,0 nl
s Spin momentum quantum number
The States of Atomic Hydrogen
The assembly of all hydrogen-atom states with the same principal quantum number n are said to form a shell. The collection of all states with the same value of the orbital angular momentum quantum number l is called a subshell.
For a certain angular momentum l, there are 2l+1 states, and consider spin there are 2(2l+1) states.
ml=-l, -(l-1), …-1, 0, 1, …,(l-1), l
12 l
)1(,2 ,1 ,0 nl
2
1sm
)1(
0
)12(2n
ln lZ
2[1 3 5 (2 1)]n 22n
)0(1 ,1 lsnstates 2
)0(2 ,2 lsn
states 2 6
)0(3 ,3 lsn2 6
)2,1,0,1,2(),2(3 lmld
10
states
Example
)1 ,0,1(),1 (2 lmlp
)1,0,1(),1(3 lmlp
ml=-l, -(l-1), …-1, 0, 1, …,(l-1), l
12 l
states
2
1sm
)1(
0
)12(2n
ln lZ
2[1 3 5 (2 1)]n 22n
The X-Ray Spectrum of Atoms
The Characteristic X-ray Spectrum
Atomic Magnetism
How to study the angular momentum properties of the atom ?
2
2r
el
L is the orbital angular momentum vector of the electron.
The component of z directionzlz L
m
e
2
L
l
2
2mr
m
e L
m
e
2
)1( llL
Lz=mlħ, ml=-l, -(l-1), ...-1, 0, +1, ..., +l
Atomic Magnetism
Lz=mlħ, ml=-l, -(l-1), ...-1, 0, +1, ..., +l
m
ehmm
m
eL
m
ellzlz
422
It can be expressed by Bohr magnetron µB
J/T10274.94
24m
ehB
Bllz m
We can express the magnetic dipole moment in terms of the Bohr magnetron
The energy of electron
,3 ,2 ,1
1
8 2220
4
n
nh
meEn
i fn nE E hv
If we were to place an atom having a magnetic dipole moment in a magnetic field , which we assume is in the z direction, the energy associated with the interaction between the atom and the magnetic field is
l
B
l lz z l B zU B B m B
That is, atoms with different values of ml have different energies in the field, which provides a way to determine their orbital angular momentum.
Atomic Magnetism
dBdW l )(
00
sin)(
dBdBW Ll
cosBL BL
The Stern-Gerlach ExperimentBU
zz BU
z
B
z
UF z
z d
d
d
d
there is no semiclassical corresponding. The full quantum mechanics gives
sz s Bm 0002319305.2sg
Atomic Magnetism
spin angular momentum
Bllz m angular momentum
Sodium atom (Z=11)
2
1 lslj
jmmJ jj ,1 ,0 ,
The Zeeman effect
Bsssz mg
Inverted population
Lasers and Laser Light
Stimulated
emissionEmission
Lasers and Laser Light
Four properties
1) Laser light is highly monochromatic.
2) Laser light is highly coherent
3) Laser light is highly directional
4) Laser light can be sharply focused.
ExercisesP1099 15, 16, 17P1100 28, 30