occurs in all directions simultaneously from the
lowest state (n = 1, n = 1, n = 1) to the excited
state (n = 2, n = 2, n = 2). What wavelength, in
nm, of radiation would the electron absorb?
Would you expect the solution to be colored?
10. Boundary conditions are important because
they describe the confinement of our particle
of interest. It is this confinement that naturally
leads to the quantization of the energy levels
of a system. By considering some of the
models described in this chapter, describe
the how quantization changes as we excite
the system to extremely high quantum states.
11. The wavefunction described by Equation 4.32
is not normalized. We can use the condition
∫ψ (x) ψ* (x) dx = 1 to normalize the
wavefunction and determine the normalization constant A.
(a) Using the fact that A
2
ð
2π
0
e
inϕ e
−inϕ dϕ = 1 and
the general relation e
a × e
b = e
(a+b) , show
A
2
ð
2π
0
dϕ = 1
(b) Integrate the above equation and show that
the normalization constant is
A =
ffiffiffiffiffi ffi
1
2π
r
References and recommended reading
Atkins, P. W. Quanta: A Handbook of Concepts, Second
ed. 1991, Oxford University Press. This is more of a nonmathematical and highly visual approach to understanding ideas in quantum mechanics. The book is full of
detailed illustrations that make grasping concepts easy.
Brus, L. E. A simple model for the ionization potential,
electron affinity, and aqueous redox potentials of small
semiconductor crystallites. The Journal of Chemical
Physics. 1983, 79: 5566–5571. This paper introduces the
application of the particle-in-a-sphere model to quantum
dots. Additional explanation and a more detailed model
can be found in Liu, H. Brozek, C. K., Sun, S., Lingerfelt,
D. B., Gamelin, D. R., and Li, X., A Hybrid QuantumClassical Model of Electrostatics in Multiply Charged
Quantum Dots. Journal of Physical Chemistry C. 2017,
121: 26086–26095.
Kuhn, H., Försterling, H.-D. and Waldeck. D. H. Principles
of Physical Chemistry, 2nd ed. 2009, John Wiley and Sons,
Inc., New Jersey. This physical textbook takes a particlein-a-box approach to understanding the essence of
quantum mechanics and why atoms and molecules exist.
Lowe, J., and Peterson, K. Quantum Chemistry, 3rd ed.
2005, Academic Press. A great introduction to basic
quantum mechanics that assumes little mathematical
or physics knowledge. The book is written to enable
students and researchers to comprehend the current
literature.
CHAPTER 4: Quantum Effects at the Nanoscale
132
lowest state (n = 1, n = 1, n = 1) to the excited
state (n = 2, n = 2, n = 2). What wavelength, in
nm, of radiation would the electron absorb?
Would you expect the solution to be colored?
10. Boundary conditions are important because
they describe the confinement of our particle
of interest. It is this confinement that naturally
leads to the quantization of the energy levels
of a system. By considering some of the
models described in this chapter, describe
the how quantization changes as we excite
the system to extremely high quantum states.
11. The wavefunction described by Equation 4.32
is not normalized. We can use the condition
∫ψ (x) ψ* (x) dx = 1 to normalize the
wavefunction and determine the normalization constant A.
(a) Using the fact that A
2
ð
2π
0
e
inϕ e
−inϕ dϕ = 1 and
the general relation e
a × e
b = e
(a+b) , show
A
2
ð
2π
0
dϕ = 1
(b) Integrate the above equation and show that
the normalization constant is
A =
ffiffiffiffiffi ffi
1
2π
r
References and recommended reading
Atkins, P. W. Quanta: A Handbook of Concepts, Second
ed. 1991, Oxford University Press. This is more of a nonmathematical and highly visual approach to understanding ideas in quantum mechanics. The book is full of
detailed illustrations that make grasping concepts easy.
Brus, L. E. A simple model for the ionization potential,
electron affinity, and aqueous redox potentials of small
semiconductor crystallites. The Journal of Chemical
Physics. 1983, 79: 5566–5571. This paper introduces the
application of the particle-in-a-sphere model to quantum
dots. Additional explanation and a more detailed model
can be found in Liu, H. Brozek, C. K., Sun, S., Lingerfelt,
D. B., Gamelin, D. R., and Li, X., A Hybrid QuantumClassical Model of Electrostatics in Multiply Charged
Quantum Dots. Journal of Physical Chemistry C. 2017,
121: 26086–26095.
Kuhn, H., Försterling, H.-D. and Waldeck. D. H. Principles
of Physical Chemistry, 2nd ed. 2009, John Wiley and Sons,
Inc., New Jersey. This physical textbook takes a particlein-a-box approach to understanding the essence of
quantum mechanics and why atoms and molecules exist.
Lowe, J., and Peterson, K. Quantum Chemistry, 3rd ed.
2005, Academic Press. A great introduction to basic
quantum mechanics that assumes little mathematical
or physics knowledge. The book is written to enable
students and researchers to comprehend the current
literature.
CHAPTER 4: Quantum Effects at the Nanoscale
132