Nanotechnology-Enabled Sensors

6.2 Density and Number of States 285
dimensional mechanical structures, as it provides the necessary tools to
understand their behaviour and describe their properties.
6.2.2 Momentum and Energy of Particles
Free particles, such as molecules in an ideal gas, do not interact significantly
with each other. For example, in a host material the valence electrons
detach themselves from the host atoms and move freely in a state
much like molecules in an ideal gas. The collective state of these electrons
is called Fermi gas.
For a system of free particles, the potential energy is zero, and the total
energy of the system is comprised of the kinetic energy of the constituent
particles. Kinetic energy is classically defined as:
E = ½mv 2 = p 2 /2m, (6.1)
where p is the momentum, m is the particle’s mass and h is the Plank’s
constant.
Louis De-Broglie was the first to hypothesize that particles could exhibit
wave-like properties. Quantum mechanics treats particles in this fashion.
In quantum mechanics, the total energy of a particle is given by the Hamiltonian
of the system, which is comprised of kinetic and potential energy
components.
To determine the momentum of a free particle using quantum mechanics,
the wave-like nature of the particle must be considered. The relationship
between frequency f and wavelength λ of a wave is:
λf = v, (6.2)
where v is the velocity of the wave. The wavevector (or wavenumber), k, is
defined as:
In classical physics, momentum p is defined as:
k = 2π/λ. (6.3)
p = mv. (6.4)
However, in quantum mechanics, matter-waves are responsible for the
propagation of momentum, which is thereby defined as:
⎛ h ⎞
p = ⎜ ⎟k
= �k
, (6.5)
⎝ 2π
⎠