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