WAVE- PARTICLE PARADOX
The wave particle duality refers to the idea that light and matter (such as electrons) have both wave and particle
properties, that is light behaves either as a wave or as a particle but not as both simultaneously.
Wave nature of matter
De-Broglie assumed that moving electrons behave like waves
The phenomena that can be rightly explained by assuming that matter (light or electrons) behaves like a wave
are:
1. Diffraction of light/electron.
2. Reflection of light/electron
3. Refraction of light/electron
4. Interference of light/electron
5. Polarization of light/electron
The electrons were diffracted as it passes through the thin metal film and the diffraction rings of electrons were
produced on a photographic plate.
If the voltage V on the anode increases, the velocity v of the electrons will increase and the rings will become
narrower. The wavelength λ of the electron decreases with increasing electron velocity.
De-Broglie suggested that electrons/lights have wave properties. He stated that the wavelength of the wave
carried with the electrons/lights was given by
h
λ= ; where h= planck constant ; p=momentum n
p
h
λ= ; where m=mass , v=velocity
mv
Device whose invention is based on the wave nature of matter includes:
- Electron microscope: It is a microscope that uses accelerated electron as a source of illumination
- Scanning tunneling microscope: It is an instrument for margin surfaces at the atomic level
Particle nature of matter
The phenomena that can be explained in term of the particle nature of light include.
1. Photo-electricity e.g X-rays fall on the surface of a thin sheet of metal, they collide with the atoms of
the metal and eject electrons.
2. Compton effect: When a single x-rays photon collides with a free electron, the electron recoil off as
though it were perfectly elastic sphere
3. Thermionic emission
4. Radiation of energy from hot bodies
5. Emission and absorption of light
6. Quantization of radiation energy proposed by Planck.
The particles used in explaining the wave nature of matter are:
- Hydrogen atom/proton
- Helium atom
- Helium nuclei/alpha particles
- Neutron
- Protons
The Uncertainty Principle
Heisenberg uncertainty principle states that it is impossible to determine accurately both the wave properties of
matter and its particulate properties at the same time. OR
Heisenberg uncertainty principle states that there is always an uncertainty in any attempt to measure the
position and momentum of a particle simultaneously or energy and time as complementary variable.
h
Δ x . Δ p≥ ; where Δ x=change ∈ position; Δ p=change∈momentum
2π
h
Δ E . Δt ≥ ; where ΔE=change∈energy ; Δt=change∈time ; h=Planck constant
2π
h
Δ x . Δv ≥ ; where Δ x=change∈ position; Δv=change invelocity
2π
Example:
- The uncertainty in determining the duration which an electron remains in a particular energy level
before returning to the ground state is 2 x 10-9 s. Calculate the uncertainty in determining its energy at
that level. [ h
2π
=1.054 ×10−34 Js
]
Solution
h
Δ E . Δt ≥
2π
h 1
ΔE≥ .
2π ∆t
−34
1.054 ×10
ΔE≥ −9
2 ×10
−26
Δ E ≥ 5.27 ×10 J
- The uncertainty in locating the position of an electron is 5 x 10-11 m. Calculate the uncertainty in
determining the momentum of the electron ( h=6.626 ×10−34 Js ; π=3.142 )
Solution
h
Δ x . Δ p≥
2π
h 1
Δ p≥ .
2π Δx
−34
6.626 ×10
Δ p≥ −11
2 ×3.142 ×5 × 10
−24
Δ p ≥ 2.11×10 Ns
- Calculate the wavelength of an electron moving at a velocity of 106 m/s. Given that (h = 6.6 x 10-34Js, Me
= 1 x 10-30 kg )
Solution
h
λ=
mv
−34
h 6.6 × 10 −10
v= = −30 6
=6.6 × 10 m
mλ 1 ×10 ×10
Calculate the velocity of an electron with wavelength of 10-15m. What will be the potential difference of
the electron before it could be accelerated to acquire that velocity. (h = 6.6 x 10-34Js, Me = 9.1 x 10-31 kg ,
1e = 1.6 x 10-19C)
Solution
h
λ=
me v
−34
h 6.6 ×10 11 −1
λ= = =7.25 ×10 m s
mv 10−15 × 9.1 ×10−31
1 2
eV = me v
2
2
me v 2 9.1 ×10−31 × ( 7.25 × 1011 ) 12
V= = =1.495 ×10 volts
2e 2 ×1.6 × 10
−19
