ch 5. the particle in the box and the real world ms310 quantum physical chemistry - understanding...

Ch 5. The Particle in the Box Ch 5. The Particle in the Box and the Real Worldand the Real World

MS310 Quantum Physical Chemistry

- Understanding the tunneling of Q.M particles - Understanding the tunneling of Q.M particles through barriers and size quantizationthrough barriers and size quantization

- Principle of the scanning tunneling microscope Principle of the scanning tunneling microscope (STM)(STM)

- Engineering of device called a quantum well Engineering of device called a quantum well structurestructure ex) Quantum well, Quantum dotsex) Quantum well, Quantum dots


5.1. The particle in the finite depth box

In the real world, the box have a finite depth

Particle ‘in’ the well : E < V0

We can divide the equation inside and outside the box.1) Inside the box : V(x)=0

)(2)(

22

2

xmE

dx

xd

2,

2for ,

22for ,0)(

0

ax

axV

ax

axV

However, wave function is not zero at the end of boxWhy? Box is ‘finite’ depth


2) Outside the box : V(x)=V0

)()(2)(

20

2

2

xEVm

dxxd

kxDkxCx cossin)(

Solution is same as infinite well

xx

xx

eBeAxa

x

EVmBeAex

ax

''

20

)(:2

ii)

)(2,)(:

2 i)

Limiting behavior : ψ(∞) = ψ(-∞) = 0 : B = A’ = 0

2for

)(2,

2for )(

'

20

axeB

EVmaxAex

x

x


Hamiltonian operator and potential is symmetry to y-axis. → we can divide the solution to even and odd functioni) Even function : A’ = B, C=0ii) Odd function : A’ = -B, D=0 Solution is given by this picture and we can see the even and odd function alternative.

Solution of V0=1.20x10-18J and width 1.00x10-9m

Outside the box : classical forbidden region.(Ekinetic<0). However, this state allowed in the Q.M

1) Solution depends on the m, a, V0 and finite number of states.

2) ψ(x) decays rapidly when V0 >> E and slowly when V0 ~ E

5.2 Difference in overlap between core and valence electrons


We take finite depth box as the crude model of atom.

We take finite depth box as the crude model of atom.

Strongly bound level : core electronsweakly bound level : valence electrons

The second atom close to first atom. Wave function of weakly bound state : ‘significant overlap’Strongly bound state : wave functions have a small overlap.


Overlap of wave functions n=1 to n=5

Except to n=5, there are almost not overlapping each other. However, when n=5, two wave functions are significant overlapping. → make a ‘bonding’


5.3 Pi electrons in conjugated molecules can be treated as moving freely in a box

Absorption of light in the UV-Vis range : electron excitation occupied level → unoccupied level

Electrons are delocalized in some organic molecules → π-bonded network, absorption spectrum shift to UV range. → Delocalization of electron

Delocalized electron : seems to move freely over the whole π-bonded network → described by 1-dimensional particle in a box

The longest wavelength of π-conjugated molecules1,4-diphenyl-1,3-butadiene : 345 nm1,6-diphenyl-1,3,5-hexatriene : 375 nm1,8-diphenyl-1,3,5,7-octatetraene : 390 nm


Calculate about the 1,6-diphenyl-1,3,5-hexatriene

HOMO(highest occupied molecular orbital) : n=3LUMO(lowest unoccupied molecular orbital) : n=4Transition : n=3 → n=4

pm892)s m 10kg)(2.998 108(9.11

)m10375)(sJ 10626.6)(34(

8

)(

8

)(-1831-

93422max

22222

mc

hnn

Em

hnna ifif


Ground state of 1,6-diphenyl-1,3,5-hexatriene : highest occupied energy level is n=3Assumption : # of molecules at n=4 state can negligible at 300KUse Boltzmann distribution and we can answer it.

J 1045.4m) 10kg)(973 1011.9(8

s)J 107(6.626

8

)( 192-1231

2-34

2

222

ma

nnhE if

In this case, quantum states of n=3 and n=4 is same(only 2 electrons can be in the each states)

47--123-

-19/

3

4

3

4 102.1])(300K)KJ 10(1.38

J 104.45[-exp

kTEeg

g

n

n

Therefore, we can say that all of molecule in the ground state.

5.4 Why does sodium conduct electricity andwhy is diamond an insulator?


We focus on the crystalline metal.First, see the 2 Na atoms.

See about the distance of 2 atoms 1) long : two potential separated, e- localized each atom 2) short : barrier lowered → e- delocalized into 2 atoms


Next, see the 1-dimensional Na crystal.

Na atoms make the periodic potential and one e- per atom is delocalized over the sample

20 million atoms into the 1.00cm box.

)J1003.6)(12(8

)12(

88

)1(

34-2

2

2

22

2

22

1

nma

hn

ma

hn

ma

hnEEE nn

it is basically energy band


Finally, we see the idealized case of periodic potential.

Red region : occupied statesPink region : unoccupied states(unfilled valence band)

Beyond the dotted lined : There are no allowed level until the energy increase by ∆E

Case of Na : valence band is ‘partially’ filled.

Therefore, e- in sodium can easily move into the box with very small(seems to continuous) energy. → conductor!


If the electric field is applied?

If the electric field applied, band energy decreased by b).It changes to c) after electron transfer.

Energy level is so narrow, wave functions are significantly overlapped.(overlap of wave functions) → easy to response to electric field(conduction!)


What difference exists between sodium and diamond?

Na : only 1 3s orbital filled → partially filled valence bandDiamond : all states in the band accessible to the delocalized valence e- are filled !Transition in insulator and semiconductor is valence band to conduction band → need high energy and difficult to electron transfer

5.5 Tunneling through a barrier


Classically, we cannot climb up the mountain if we don’t have enough energy to go to highest of the mountain.

However, what about the Q.M barrier? → particle ‘smear out’ the barrier although not enough energy : ‘Tunneling’

Potential is described by

V(x) = 0 for x < 0 – (1) = V0 for 0 ≤ x ≤ a – (2) = 0 for x > a – (3)


ikxikx BeAex

mEkmE

kxkxmE

dx

xdxE

dx

xd

m

)(

))2(,2

)(()(2)(

),()(

2

1

222

22

2

2

2221

Region (1) : x < 0, V=0

Region (2) : 0 < x < a, V=V0, E - V0 < 0

))2(),(2

()(

)()(2)(

),()()()(

2

21

022

2

2022

2

2

22

mEkEVm

DeCex

xEVm

dx

xdxExxV

dx

xd

m

xx

Region (3) : x > a, V=0

ikxikx eBeAx

mEkmE

kxkxmE

dx

xdxE

dx

xd

m

''3

222

22

2

2

22

)(

))2(,2

)(()(2)(

),()(

221

ikaikaaa

ikaikaaL

eikBeikADeCe

DCikBikA

eBeADeCe

DCBADeCeBeAe

dx

ad

dx

adaaax

dx

d

dx

dx

''

''

0000

3222

2121

i.e.

)()(,)()(at

)0()0(,)0()0(0at

In region (III), no reflected wavefunction expected : A’=0

ax

xA

x

xA

of region the in

direction to travelling particle a of yprobabilit The :

0 of region the in

direction in travelling being particle a of yprobabilit The :

2'

2

Wavefunction must satisfy the continuity.


Wave function decays into the barrier : e-κx term1/κ : decay lengthProbability of high-energy particle > probability of low-energy particle

Transmission probability is given by )(

)1(16)(

1

1

02

2'

V

E

eeA

AT aa

E < V0

E > V0

21

)2( 0mVa

aaa e

eeT

22 )1(16

)1(16

)(1

1

0

If κa >> 1 condition(high, wide barrier)

5.6 The scanning tunneling microscope(STM)


Gerd Binning & Heinrich Rohber : invent the STM and get Nobel Price in 1986

Typically, electrons with energy of 5eV can tunnel from the metal tip to the surface.

Electron energy → work function(E-V0), decay length of electron : about 0.1nm

External voltage(0.01V-1V) applied(see picture c)) to obtain tunnelingIf d(surf-tip)is about 0.5nm → decreases the order of magnitude of tunneling current when barrier increases each 0.1nm : ‘microscope’


a) Initial state : electrically separated between measured surface and scanner

b) Measured surface and scanner connected by external circuit → ‘same potential’

c) External voltage applied and generated the potential barrier eV → e- tunneling left to right


Whole device of STM


Electrical current of tunneling : exponentially decreaseIn this case, V is constant.I decrease exponentially → R increase exponentially → AFM(Atomic Force Microscopy)AFM : use gold substrate and thiol.

5.7 Tunneling in chemical reactions


Reactant must over the barrier, activation energy.Classical view : reactant must have ‘enough energy’ to over the activation energy.However, it can by the tunneling.

5.8 Quantum wells and quantum dots


Quantum well : energy is confined into 2-dimensionExample of quantum well : GaAs layer, AlαGa1- α As layerWe can make heterostructures by alternating GaAs layer and AlαGa1- α As layer.

Semiconductor : fully occupied valence band and band gap between valence band and conduction band

band structure of GaAs layer, AlαGa1- α As layer


We make this structure.

Technically, height of layer can be 0.1 to 1nm and width is order of 1000nm scale.

Energy is given by

Therefore, energy is essentially continuous to ny and nz, but discrete to nx.

order] 1000nm:b 10nm,-1:[a)(8 2

2

2

222

a

n

b

nn

m

hE xyz

nnn zyx


Ground state : fully occupied valence bandIn this case, the lowest excitation occurs in GaAs layer.(the highest level electron)If putting E larger than gap of GaAs, smaller than that of AlαGa1- α As → photon is emitted with frequency ν=∆E/h when system decays to the ground state(∆E:band gap)

Lasers use the quantum well, there are 2 advantages.1) very efficient in producing photons

2) ∆E can be changed easily by the thickness of the layer, following the equation

This technique : Molecular Beam Epitaxy(MBE) → expensiveNew technique : Quantum Dot

)(8 2

2

2

222

a

n

b

nn

m

hE xyz

nnn zyx

Quantum dots : nanometer scale , energy is confined into 3-dimension

Three-dimensional nanocrystals of

semiconducting materials containing 103

to 105 atoms

Ex) CdSe nanoparticles

2

22

8 Lm

hnE

en

<TEM image of Quantum Dots>

Energy required to create mobile charge carrier (electrical

conductivity) and create electron-hole pair (e—hole recombination :

optical property)

→Depends on the size of the quantum dot


Make a aqueous quantum dot : attach the hydrophilic group

Size → energy gap → λem

Absorption and PL spectra forseveral sizes of InP nanocrystals

A. A. Guzelian et al., J. Chem. Phys.

(1996)

Different emission with different quantum dot sizes

Quantum dots array of InAs on the GaAs substrate

(Mirin et al. 1996)

1 μm

1 μm

Example of quantum dot


- The tunneling of quantum mechanical particles through barriers and size quantization was discussed.

- The understanding of why chemical bonds involve the least strongly bound, or valence, electrons and not the more strongly bound, or core, electrons was provided.

- To investigate tunneling, It focus on the region near the edge of the finite depth box modified by making it wider and letting the barrier having a finite thickness on the right-hand side.

- It is useful for engineering of the device called a quantum well structure such as quantum well and quantum dots.

Summary Summary

ch 5. the particle in the box and the real world ms310 quantum physical chemistry - understanding...

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