Post on 13-Jun-2022
Chapter 2
Free particle:2 2 2
2 2 2
22
d d mEEm dx dx
ψ ψψ ψ−= → = −
k
kx
i
i
xA e
A eψ
ψ+
−− −
+
=
=
22 /k mE=
Note: x can take on any value, but px is eitheror (consistent with uncertainty principle)
*( )2*
L
L
dxp x dxLdx
ψ ψ
ψ ψ−
= =∫
independent of x.
traveling wave
traveling wave
k k−
Equal probability of finding the particle anywhere
L →∞ in the case of a free particle
(V ≡ 0) c o s s i n
1
, 2 /
i k x
i k x
e k x i k x
e
p k khp
λ π
λ
±
±
= ±
=
= =
=
2
' '
2
2 /
' 2 ( ) /
' ( real)
ikx ikxI
ik x ik xII
x xII
I Ae Be
k mE
II Ce De
k m E V
E Vk i
Ce De
ψ
ψ
ψ
−
−
−
= +
=
= +
= −
<=
= +k k
k k
For proper wavefunction, D = 0
V
I
O x
II
2 2
2
2 2
2
2
2
I
II
dHm dx
dH Vm dx
−=
−= +
Particle incident on a step potential
ψ exponentially decaying in region II
Transmission, T = # particles transmitted
# particles incident
Reflection, R =# particles reflected
# particles incident
( )
( )
2
2 2
2
2 2
4 ''
4 '1'
'Ckk kTkk k A
BkkRk k A
= =+
= − =+
1R T+ =
(0) (0)
' (0) ' (0)
I II
I II
A B C
Aik Bik C
ψ ψ
ψ ψ
= ⇒ + =
= ⇒ − = − k
If E > V
Barrier of finite width
' '
'' ''
, 2
' ' , ' 2 ( )
'' '' , '' 2
ikx ikxI
i x ik xII
ik x ik xIII
Ae Be k mE
A e B e k h m E V
A e B e k mE
ψ
ψ
ψ
−
−
−
= + =
= + = −
= + =
II
V
III
0 L x
( )
2 2
2 2
'' 1, 1 16 1
1 ~1
16
L l
L L
B AR T
E EA A e eV V
E V Te e
−
−
= = =⎡ ⎤⎛ ⎞+ − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
<<−
+
k k
k k
E < V, assume particle incident from left
'k i= k
I
22 2 ( ) /oL m V Ee− −T~
resonances:particle canescape bytunneling
resonance
bound
Examples: • radioactive decay
• temporary anions:Be-, N2
-, benzene-
electron falls offin ≈10-14 sec V(r)
How can one measure something with such a short lifetime?
E
r
Scanning tunneling microscopy (STM) – invented ~20 years ago at IBMResearch Labs, Zürich
adsorbatesurface
Apply voltage – measure current
often run so that as the tip is scanned overthe surface, the height is varied so as to keepthe current constant
the tip does not actually touch the surface
electrons tunnel between tip and surface
tip
Water chains on the Cu(110) surface, from Yates et al.
Particle the 1-D boxparticle cannot escape from the box
Inside the box2 2
22d E
m dxψ ψ−=
( ) sin cos
(0) 0 sin cos 0
( ) sin
( ) 0 sin , 1, 2,3,...
2( ) sin sinn
x D kx C kx
D kx C kx C
x D kx
L D kL kL n n
n x n xx DL L L
ψ
ψ
ψ
ψ π
π πψ
= +
= = + ⇒ =
=
= = ⇒ = =
= = normalized
ApplyBoundaryConditions(0) ( ) 0Lψ ψ= =
Wavefunction inside box is:
L0
2 2
2
2 2 2
2
2 2 2 2 2
2 2
sin sin2
( 1)2
, 1,2,3...2 8n
d n x n xEm dx L L
n Em L
n n hE nmL mL
π π
π
π
− ⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
−− =
= = =
minimum energy = = zero-point energy
Consistent with the uncertainty principle.
2
28hmL
Because x is constrained to bebetween 0 and L, the momentumcannot be zero. E ≠ 0.
for all .2
< 0 for all .x
Lx n
p n
< > =
> =
Energies get closer together asm → ∞L → ∞
2
1 2 (2 1)8n n
hE E E nmL+Δ = − = +
Excitation energy
⇒
L
ethyleneH
C = CH
H
H1.5 2(.5) 2.5 L ≈ + = Å
butadieneH
C = CH
H
C = C
H
H
H
3(1.5) 2(.5) 5.5L ≈ + = Å
hexatrieneH
HC = C
H
C = CH
H
C = C
H
H
H
5(1.5) 2(.5) 8.5 L ≈ + = Å
( )L a≡
Can use as a crude model for understanding the electronic spectraof polyenes.
ethylene: 2 π electrons ΔE: n = 1 → n = 2
butadiene: 4 π electrons ΔE: n = 2 → n = 3
hexatriene: 6 π electrons ΔE: n = 3 → n = 4
2
2 15.4 eV8
hmL
=
2
2 3.2 eV8
hmL
=
2
2 1.3 eV8
hmL
=
π orbitals
butadieneethylene
UV3.1 eV(400 nm)
violetred
1.8 eV(700 nm)
IR
HOMO LUMO
HOMO LUMO
tunneling
Particle in a finite box
Finite # of bound levels
Tunneling into classically forbidden regions