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Nanoelectronics
Chapter 2 Classical Particles, Classical
Waves, and Quantum Particles
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Electron Double-Slit Experiment
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2.1 Comparison of Classical and
Quantum Systems
• Classical physics is to describe the exact state
of a particle, how fast it will travel at a certain
instant of time.
• Quantum mechanics: it is impossible to
measure precisely both the position and
momentum of a particle, theoretically
impossible.
• Quantum theory is truly a probabilistic theory.
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2.2 Origins of Quantum Mechanics
• Photoelectric effect: if light is incident on a
metal, some energy carried by the light can be
transferred to electrons at the metal’s surface,
and the electrons gain enough energy to
escape from the metal.
• Kinetic energy of photo-emitted electrons
does not increase as the intensity of light
increases.
• When the frequency of light increases, more
energetic electrons were emitted.
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2.3 Light as a Wave, Light as a Particle
• Light as a particle or perhaps a wave – the
early years
• A little later – Light as a wave
– Interference
Ψ𝑇 = Ψ1 + Ψ2 = 𝑒 𝑖𝑘𝑑 + 𝑒 𝑖𝑘(𝐿+𝑑) = 𝑒 𝑖𝑘𝑑 (1 + 𝑒 𝑖𝑘𝐿 )
If kL = 0, 2π, 4π, … Ψ𝑇 = 2
If kL = π, 3π, 5π, … Ψ𝑇 =0
– Young’s Experiment – What to expect from Waves
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Young’s Experiment
Ψ𝑇
𝑒 𝑖𝑘𝑟1
𝑒 𝑖𝑘𝑟2
= 𝐴(
+
)
𝑟1 from jth𝑟slit
2 to
r is the distance
j
observation location on the screen
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Finally, Light as a Quantum Particle
• Young’s Experiment – one slit at a time
|𝜓1 |2 + |𝜓2 |2 ≠ |𝜓1 + 𝜓2 |2
|𝜓1 + 𝜓2 |2 = |𝜓1 |2 + |𝜓2 |2 + 2𝑅𝑒(𝜓1 ∗ 𝜓2 )
• Young’s Experiment – What to expect with
classical particles
𝐼𝑇 = 𝐼1 + 𝐼2
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Finally, Light as a Quantum Particle
• Young’s Experiment and the Concept of
Photons – one more time with light, but
slowly.
If the intensity is reduced enough, we would find that
energy is not arriving continuously, but in discrete
bursts, pointing to a particle-like nature of light.
Photon:
𝐸 = ℎ𝑓 = ℏ𝜔
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Finally, Light as a Quantum Particle
• Young’s Experiment – A very strange result
concerning interference
Light exhibits bot wave-like and particle-like
behavior, is clearly neither a classical wave not a
classical particle. We call this a quantum
particle.
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2.4 Electrons as Particles,
Electrons as Waves
• 2.4.1 Electrons as Particles - The Early Years
It was found in late 1800s by J.J. Thomson.
• 2.4.2 Electrons (and everything else) as Quantum
Particles
Louis de Broglie in 1923 suggested that all
“particles” having energy E and momentum p
should have wavelike properties, too.
ℎ
𝑝
𝜆 = ; 𝑝 = ℏ𝑘; k is wavenumber,
- matter waves, for 1Kg object: 𝜆 = 6.6 × 10−34 𝑚
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Electrons as Waves
• The wavelength of a 1 eV photon:
ℎ𝑐
𝜆𝑝 =
= 1.24 𝜇𝑚
𝐸
• The wavelength of a 1 eV electron:
ℎ
𝜆𝑒 =
= 1.23 𝑛𝑚
2𝑚𝑒 𝐸
• Why we cannot see individual photon or the
granularity of light flow?
1 uW light has 1012 photon/s
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2.4.3 Further Development of
Quantum Mechanics
• Schrodinger’s equation
• (In double-slit experiment) the wave 𝜓 of each
photon interferes with itself in passing through
the two slits. Or the photon pass both slits at the
same time
• Heisenberg uncertainty principle Δ𝑝Δ𝑥 ≥ ℏ/2
• Particles with integral spin: bosons, Particles with
half-integral spin: fermions
• Pauli exclusion principle: two or more identical
fermions cannot occupy the same quantum state.
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2.5 Wavepackets and Uncertainty
• For a particle with mass m: 𝐸 = ℏ𝜔 =
• Dispersion relations: 𝜔(𝑘) =
• Phase velocity: 𝑣𝑝 =
ℏ𝑘 2
2𝑚
ℏ𝑘 2
2𝑚
𝜔
𝑘
• It seems reasonable to model a quantum particle
as being associated with a wavepacket, since a
wavepacket exhibits wave-like behavior, viewed
like a particles from a far distance. Δ𝑘 Δ𝑥 = 𝜋/2
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2.5 Wavepackets and Uncertainty
• Δ𝑘 Δ𝑥 = 𝜋/2
• A wavepacket tightly confined in space
Is made up of plane waves having a large
Spread in wavenumbers, vise versa.
• Group velocity: 𝑣𝑔 =
𝜕𝜔
|𝑘=𝑘0
𝜕𝑘
Wavepacket
can change
𝜋
shape
∆𝑥 = 𝑣𝑔 𝑡 −
2∆𝑘
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2.6 Main Points
•
•
•
•
•
•
Classical particles
Quantum particle’s position and momentum
Electron’s wavelength
Idea of spin and Pauli exclusion principle
Fermions and bosons
Wavepacket concepts: phase velocity and
group velocity
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2.7 Problems
• 2, 3, 4, 11, 12
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