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1. Applied Modern Physics
EE1104

2. Lecture -4
Wave Properties of Particles

3. De Broglie Waves (Matter Waves)
• Background
• Before 1924, wave–particle duality was known only
for light:
• Light behaves as a wave (interference, diffraction).
• Light behaves as a particle (photoelectric effect,
photons).
2

4. Cont’d…
• Louis de Broglie (1924) proposed that all moving
particles (like electrons, protons) also exhibit wave-like
behavior.
• This idea unified the concepts of waves and particles.
• From the following figure and mathematical equation,
one of the physical meaning is
− A particle’s position is not fixed; it is described by a
probability wave that spreads out in space.
3

5. Cont’d...
• Any moving particle (like electrons, protons) has an
associated wavelength, called the de Broglie
wavelength.
4

6. Cont’d…
• Derivation of de Broglie Equation:
- according to Einstein: 𝐸 = 𝑚𝑐2
……………………...(1)
- according to Planck’s Quantum theory: 𝐸 = ℎ𝑓 …..(2)
𝑚𝑐2 = ℎ𝑓
𝑚𝑐2 = ℎ
𝑐
𝜆
𝑚c =
ℎ
𝜆
⇒ 𝝀 =
𝒉
𝒎𝒄
=
𝒉
𝒑
… … . . . . . . . . . . . . . . . . . . . . 3
Equation (3) is deBroglie Equation (deBroglie wave length)
5

7. Cont’d…
• Where:
− λ = de Broglie wavelength
− ℎ = Planck’s constant (6.626 × 10⁻³⁴ J·s)
− 𝑝 = 𝑚c = mv = momentum of the particle
• 𝝀 => relates to a wave like property
• 𝒑 => relates a particle like property
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8. Phase and Group Velocity
• A wave can be described as a combination
(superposition) of many small waves.
• Each wave travels with a certain phase velocity, and
together they form a wave packet that moves with a
group velocity.
7

9. Cont’d…
• Phase Velocity (vₚ):
• The phase velocity is the speed at which a point of
constant phase (like a crest or trough) travels.
• Mathematically: 𝑣𝑝 =
𝜔
𝑘
− Where: 𝜔 = angular frequency, 𝑘 = wave number.
• For a de Broglie wave, 𝑣𝑝 =
𝐸
𝑝
8

10. Cont’d…
9
• Using Einstein’s relation 𝐸 = 𝑚𝑐2
𝑎𝑛𝑑 𝑝 = 𝑚𝑣
𝑣𝑝 =
𝑐2
𝑣
⇒ It can be greater than the speed of light, but no
information or energy travels at this speed — so it doesn’t
violate relativity.

11. Cont’d…
• Group Velocity (𝑽𝒈)
• The group velocity is the speed at which the overall
wave packet (and hence the particle itself) moves.
• It represents the velocity of energy or information
transfer.
• Mathematically: 𝑣𝑔 =
𝑑𝜔
𝑑𝑘
10

12. Cont’d…
• For de Broglie waves: 𝑣𝑔 = 𝑣
⇒ The group velocity equals the actual velocity of the
particle.
• Relationship Between Them:
− For a free particle: 𝑣𝑝𝑣𝑔 = 𝑐2
− That means if one is large, the other is small.
11

13. Cont’d…
12

14. Cont’d…
• Example:
13

15. Cont’d…
• Exercise:
1. an electron is accelerated using a potential difference of
120 V. calc. the de Broglie wavelength of the electron.
2. Find the wave length of a 0.057 kg tennis ball traveling
at 120 km/h
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16. Wave-Particle Duality
• Wave-Particle Duality states that every particle or
quantum entity (like electrons, photons, etc.) exhibits
both wave-like and particle-like properties.
• Wave Nature: Particles such as light and electrons can
show interference and diffraction, phenomena typically
associated with waves.
• Example: In the double-slit experiment, when light or
electrons pass through two narrow slits, they create an
interference pattern on a screen — a hallmark of waves.
15

17. Cont’d…
• Particle Nature: These same entities also behave like
particles, showing discrete packets of energy or
momentum.
• Example: In the photoelectric effect, light hitting a
metal surface ejects electrons, but only if its frequency
is high enough — indicating that light consists of
energy quanta called photons.
16

18. Cont’d…
• The de Broglie hypothesis (1924) extended this duality
to matter: 𝜆 =
ℎ
𝑝
• This means all matter — even a moving baseball or
human — has a wavelength, though it’s only noticeable
at atomic scales.
17

19. Cont’d…
18

20. Cont’d…
19
• Modern Understanding
− Wave-particle duality isn’t about switching between
“wave” and “particle” forms.
− Instead, quantum objects are described by a wave
function — a mathematical entity that gives
probabilities for finding a particle in a particular
place or state.

21. Probability and the Wave Function
20
• Wave Function (ψ): The wave function, usually
denoted by the Greek letter ψ (psi), is a mathematical
description of the quantum state of a particle or
system.
• It contains all the information about the particle —
such as its position, momentum, and energy — but in a
probabilistic form.

22. Cont’d…
• The wave function itself is not directly observable.
• However, the square of its magnitude gives the
probability density of finding the particle in a
particular region.
21

23. Cont’d…
22

24. Uncertainty Principle and Its Applications
• Proposed by Werner Heisenberg (1927), the
Uncertainty Principle states that it is impossible to
simultaneously determine both the exact position and
the exact momentum of a particle.
• Mathematically, this can be expressed as:
23

25. Cont’d…
• Where:
− Δx: uncertainty in position
− Δp: uncertainty in momentum
− h: Planck’s constant (≈ 6.626×10−34 Js)
• This means: If you know the position very accurately
(small Δ𝑥), the momentum becomes very uncertain
(large Δ𝑝), and vice versa.
24

26. Cont’d…
• Implications of the Principle
− Limitations of Measurement: The principle implies
that the more precisely we know a particle's position
(Δx is small), the less precisely we can know its
momentum (Δp becomes large), and vice versa.
• The uncertainty principle is not due to limitations of
measuring instruments. it is a fundamental property of
nature.
25

27. Cont’d…
• Example: an electron is traveling a long the x-axis with
a uniform velocity of 2𝑥106 𝑚
𝑠
. If the velocity has been
calculated to a precision of 0.15%, calculate the max.
accuracy with which the position of the electron could
be measured at the same instant in time. (assume 𝑚𝑒 =
9.11𝑥10−31𝑘𝑔)
26

28. Cont’d…
• let’s work it through step-by-step.
27

29. Cont’d…
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− Answer (interpreted):The maximum accuracy (i.e.
smallest possible uncertainty) with which the
electron’s position can be measured simultaneously
is about
• So you cannot localize the electron any better than
roughly 20 nm under the given velocity precision.

30. Cont’d…
End !
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