We will discuss light–matter interactions further in Chapters 5 and 6,
especially in the context of spectroscopic methods used to characterize
nanomaterials.
4.2.2 Matter waves and the uncertainty principle
There are two ideas central to quantum theory worth discussing. The first
is rooted in observations that very small moving particles like electrons
have wavelike properties. In other words, a particle moving with a certain
momentum (a particle-like property) will undergo wave phenomena such
as diffraction and interference. In fact, if the moving particle is small
enough it will have a measureable wavelength. This wave-particle duality
is described by the de Broglie relationship (Equation 4.4):
l =
h
p
(4.4)
In the above equation, p is the particle’s momentum, which is given by
the product of the particle’s mass (m) and velocity (v). Thus, a moving
particle with momentum p will have a corresponding de Broglie wavelength, l.
Example 4.1 The de Broglie Wavelength of a Nanoparticle
Calculate the de Broglie wavelength of a 10-ng particle moving at a
speed of 2.0 × 10
5 ms
−1
.
Solution The particle’s momentum is m × v = (1.0 × 10
−11 kg) ×
(2.0 × 10
5 ms
−1 ) = 2.0 × 10
−6 kg ms
−1 . The corresponding wavelength is
λ =
h
p
=
6:626 Â 10
−34 Js
2:0 Â 10
−6 kg ms
−1
= 3:313 Â 10
−28 m
This wavelength is too small to be measured (the diameter of a
proton is on the order of 10
−15 m). Thus, a fast moving nanoparticle
will not exhibit appreciable wavelike behavior.
The second central idea in quantum theory is the Heisenberg uncertainty principle. This principal states that it is not possible to determine
both the position and momentum of a small particle with infinite precision. Let’s consider a stationary electron. We know that its momentum is
zero but in order to locate its position we need to observe it. One way of
observing it is to use a stream of photons to locate its position. However,
CHAPTER 4: Quantum Effects at the Nanoscale
98
especially in the context of spectroscopic methods used to characterize
nanomaterials.
4.2.2 Matter waves and the uncertainty principle
There are two ideas central to quantum theory worth discussing. The first
is rooted in observations that very small moving particles like electrons
have wavelike properties. In other words, a particle moving with a certain
momentum (a particle-like property) will undergo wave phenomena such
as diffraction and interference. In fact, if the moving particle is small
enough it will have a measureable wavelength. This wave-particle duality
is described by the de Broglie relationship (Equation 4.4):
l =
h
p
(4.4)
In the above equation, p is the particle’s momentum, which is given by
the product of the particle’s mass (m) and velocity (v). Thus, a moving
particle with momentum p will have a corresponding de Broglie wavelength, l.
Example 4.1 The de Broglie Wavelength of a Nanoparticle
Calculate the de Broglie wavelength of a 10-ng particle moving at a
speed of 2.0 × 10
5 ms
−1
.
Solution The particle’s momentum is m × v = (1.0 × 10
−11 kg) ×
(2.0 × 10
5 ms
−1 ) = 2.0 × 10
−6 kg ms
−1 . The corresponding wavelength is
λ =
h
p
=
6:626 Â 10
−34 Js
2:0 Â 10
−6 kg ms
−1
= 3:313 Â 10
−28 m
This wavelength is too small to be measured (the diameter of a
proton is on the order of 10
−15 m). Thus, a fast moving nanoparticle
will not exhibit appreciable wavelike behavior.
The second central idea in quantum theory is the Heisenberg uncertainty principle. This principal states that it is not possible to determine
both the position and momentum of a small particle with infinite precision. Let’s consider a stationary electron. We know that its momentum is
zero but in order to locate its position we need to observe it. One way of
observing it is to use a stream of photons to locate its position. However,
CHAPTER 4: Quantum Effects at the Nanoscale
98