1. Ch 5. The Particle in the Box and the Real World
- Understanding the tunneling of Q.M particles through barriers and size quantization
Principle of the scanning tunneling microscope (STM)
Engineering of device called a quantum well structure
ex) Quantum well, Quantum dots
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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
However, wave function is not zero at the end of box
Why? Box is ‘finite’ depth
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Solution is same as infinite well
2) Outside the box : V(x)=V0
Limiting behavior : ψ(∞) = ψ(-∞) = 0 : B = A’ = 0
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Hamiltonian operator and potential is symmetry to y-axis.
→ we can divide the solution to even and odd function
i) Even function : A’ = B, C=0
ii) Odd function : A’ = -B, D=0
Solution is given by this picture and we can see the even and odd function alternative.
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
Solution of V0=1.20x10-18J and width 1.00x10-9m
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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 electrons
weakly 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.
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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’
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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 molecules
1,4-diphenyl-1,3-butadiene : 345 nm
1,6-diphenyl-1,3,5-hexatriene : 375 nm
1,8-diphenyl-1,3,5,7-octatetraene : 390 nm
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Calculate about the 1,6-diphenyl-1,3,5-hexatriene
HOMO(highest occupied molecular orbital) : n=3
LUMO(lowest unoccupied molecular orbital) : n=4
Transition : n=3 → n=4
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Ground state of 1,6-diphenyl-1,3,5-hexatriene : highest occupied energy level is n=3
Assumption : # of molecules at n=4 state can negligible at 300K
Use Boltzmann distribution and we can answer it.
In this case, quantum states of n=3 and n=4 is same
(only 2 electrons can be in the each states)
Therefore, we can say that all of molecule in the ground state.
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5.4 Why does sodium conduct electricity and
why 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
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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.
it is basically energy band
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Finally, we see the idealized case of periodic potential.
Red region : occupied states
Pink 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!
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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!)
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What difference exists between sodium and diamond?
Na : only 1 3s orbital filled → partially filled valence band
Diamond : 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
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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)
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Region (1) : x < 0, V=0
Region (2) : 0 < x < a, V=V0, E - V0 < 0
Region (3) : x > a, V=0
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18. Wavefunction must satisfy the continuity.
In region (III), no reflected wavefunction expected : A’=0

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Wave function decays into the barrier : e-κx term
1/κ : decay length
Probability of high-energy particle > probability of low-energy particle
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20. Transmission probability is given by
If κa >> 1 condition(high, wide barrier)
E < V0
E > V0

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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 tunneling
If d(surf-tip)is about 0.5nm
→ decreases the order of magnitude of tunneling current when barrier increases each 0.1nm : ‘microscope’
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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
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Whole device of STM
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Electrical current of tunneling : exponentially decrease
In this case, V is constant.
I decrease exponentially → R increase exponentially
→ AFM(Atomic Force Microscopy)
AFM : use gold substrate and thiol.
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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.
MS310 Quantum Physical Chemistry

26. band structure of GaAs layer, AlαGa1- α As layer
5.8 Quantum wells and quantum dots
Quantum well : energy is confined into 2-dimension
Example of quantum well : GaAs layer, AlαGa1- α As layer
We 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
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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.
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Ground state : fully occupied valence band
In 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) → expensive
New technique : Quantum Dot
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29. Ex) CdSe nanoparticles
Quantum dots : nanometer scale , energy is confined into 3-dimension
<TEM image of Quantum Dots>
Three-dimensional nanocrystals of semiconducting materials containing 103 to 105 atoms
Ex) CdSe nanoparticles
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

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Make a aqueous quantum dot : attach the hydrophilic group
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31. Example of quantum dot
Quantum dots array of InAs on the GaAs substrate
Absorption and PL spectra for
several sizes of InP nanocrystals
1 μm
1 μm
(Mirin et al. 1996)
Different emission with
different quantum dot sizes
A. A. Guzelian et al., J. Chem. Phys. (1996)
Size  → energy gap  → λem 

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Summary
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.
MS310 Quantum Physical Chemistry