2- Particle in a 2-D Rectangular Box
Schrodinger Equation is
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(
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Inside the box V=0 then
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This equation has the following solution
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(
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)
𝐸𝑛
𝑛
(
𝑚 𝐿
𝑛
𝑛
)
𝐿
1
(
𝑚 𝐿
𝐸
1
)
𝐿
𝐸
(
4
𝑚 𝐿
1
)
𝐿
𝐸
4
𝑚 𝐿
4
)
𝐿
(
Degeneracy
Two (or more) different states with the same energy = Degenerate
𝐸𝑛
𝑛
𝑚
(
𝑛
𝑛
𝐿⬚
𝐸𝑛
)
𝑛
Degenerate states –same energy
Different wave functions related by symmetry
Degeneracy = 2
Nodal Planes:
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𝑛
(
𝑚 𝐿
𝑛
)
𝐿
 Examples:
 Draw the following wave functions and give their energies
 As a variant on the free-electron model applied to benzene,
assume that the six  electrons are delocalized within a square
plate of side L=4.2 Å. Calculate the 1st π→ π* transition and
compare it to experimental value 200 nm.
 Azulene C10H8, shown below, is an aromatic hydrocarbon
containing delocalized π electrons. As a model for this π electron system, consider the mobile electrons in a rectangular
two-dimensional box of dimensions 5.00 Å by 4.65 Å.
1- How many π -electrons does azulene have?
2- Identify the quantum numbers of the HOMO and LUMO of
the π electron system
3- Calculate the wavelength (in nm) of the lowest-energy π
electron transition
4- Should azulene be a colored compound?
 Naphthalene C10H8, shown below, is an aromatic hydrocarbon
containing delocalized π electrons. As a model for this π electron system, consider the mobile electrons in a rectangular
two-dimensional box of dimensions 5.60 Å by 4.40 Å.
1- How many π -electrons does naphthalene have?
2- Identify the quantum numbers of the HOMO and LUMO of
the π electron system
3- Calculate the wavelength (in nm) of the lowest-energy π
electron transition
4- Should naphthalene be a colored compound?
3-Particle in a 3-D Box
Schrodinger Equation is
(
)
(
)
(
(
) (
)
)
Inside the box V=0 then
(
)
(
)
(
)
(
This equation has the following solution
(
)
(
)
(
(
)
(
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)
Degeneracy of Particle in a 3-D Box
 Two (or more) different states with the same energy = Degenerate
(
)
 In the Cubic box
(
)
 The lowest level E111 is non-degenerate but the second level is
threefold degenerate
 The degeneracy occurs in the symmetric cube while in the general
rectangular box the degeneracy is destroyed because the symmetry
is destroyed by making the sides with different lengths.
 Examples:
 Assume that a nucleus can be represented as a cubic box of side
10-14 m. The particles in this box are the nucleons (protons and
neutrons). Calculate the lowest allowed energy of a nucleon.
Express your result in MeV (1 MeV = 106eV, Mn = 1.67 x 10-27
kg).
 Consider the hypothetical reaction of two "cube-atoms" to form a
"moly-box". Each cube-atom contains one electron. The
interaction between electrons can be neglected. Determine the
energy change in the reaction. Comment in your results.