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Introduction

In chapters [link] and [link] the so-called wave-particle duality of light is described. This duality states that light displays properties of both waves and of particles, depending on the experiment performed. For example, interference and diffraction of light are properties of its wave nature, while the photoelectric effect is a property of its particle nature. In fact we call a particle of light a photon.

Hopefully you have realised that nature loves symmetry. So, if light which was originally believed to be a wave also has a particle nature, then perhaps particles, also display a wave nature. In other words matter which we originally thought of as particles may also display a wave-particle duality.

De broglie wavelength

Einstein showed that for a photon, its momentum, p , is equal to its energy, E divided by the speed of light, c :

p = E c .

The energy of the photon can also be expressed in terms of the wavelength of the light, λ :

E = h c λ ,

where h is Planck's constant. Combining these two equations we find that the the momentum of the photon is related to its wavelength

p = h c c λ = h λ ,

or equivalently

λ = h p .

In 1923, Louis de Broglie proposed that this equation not only holds for photons, but also holds for particles of matter. This is known as the de Broglie hypothesis.

De Broglie Hypothesis

A particle of mass m moving with velocity v has a wavelength λ related to is momentum p = m v by

λ = h p = h m v

This wavelength, λ , is known as the de Broglie wavelength of the particle (where h is Planck's constant).

Since the value of Planck's constant is incredibly small h = 6 . 63 × 10 - 34 J · s , the wavelike nature of everyday objects is not really observable.

Interesting fact

The de Broglie hypothesis was proposed by French physicist Louis de Broglie (15 August 1892 – 19 March 1987) in 1923 in his PhD thesis. He was awarded the Nobel Prize for Physics in 1929 for this work, which made him the first person to receive a Nobel Prize on a PhD thesis.

A cricket ball has a mass of 0 , 150 kg and is bowled towards a bowler at 40 m · s - 1 . Calculate the de Broglie wavelength of the cricket ball?

  1. We are required to calculate the de Broglie wavelength of a cricket ball given its mass and speed. We can do this by using:

    λ = h m v
  2. We are given:

    • The mass of the cricket ball m = 0 , 150 kg
    • The velocity of the cricket ball v = 40 m · s - 1

    and we know:

    • Planck's constant h = 6 , 63 × 10 - 34 J · s
  3. λ = h m v = 6 , 63 × 10 - 34 J · s ( 0 , 150 kg ) ( 40 m · s - 1 ) = 1 , 11 × 10 - 34 m
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This wavelength is considerably smaller than the diameter of a proton which is approximately 10 - 15 m . Hence the wave-like properties of this cricket ball are too small to be observed.

Calculate the de Broglie wavelength of an electron moving at 40 m · s - 1 .

  1. We are required to calculate the de Broglie wavelength of an electron given its speed. We can do this by using:

    λ = h m v
  2. We are given:

    • The velocity of the electron v = 40 m · s - 1

    and we know:

    • The mass of the electron m e = 9 , 11 × 10 - 31 kg
    • Planck's constant h = 6 , 63 × 10 - 34 J · s
  3. λ = h m v = 6 , 63 × 10 - 34 J · s ( 9 , 11 × 10 - 31 kg ) ( 40 m · s - 1 ) = 1 , 82 × 10 - 5 m = 0 , 0182 mm
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Although the electron and cricket ball in the two previous examples are travelling at the same velocity the de Broglie wavelength of the electron is much larger than that of the cricket ball. This is because the wavelength is inversely proportional to the mass of the particle.

Calculate the de Broglie wavelength of a electron moving at 3 × 10 5 m · s - 1 . ( 1 1000 of the speed of light.)

  1. We are required to calculate the de Broglie wavelength of an electron given its speed. We can do this by using:

    λ = h m v
  2. We are given:

    • The velocity of the electron v = 3 × 10 5 m · s - 1

    and we know:

    • The mass of the electron m = 9 , 11 × 10 - 31 kg
    • Planck's constant h = 6 , 63 × 10 - 34 J · s
  3. λ = h m v = 6 , 63 × 10 - 34 J · s ( 9 , 11 × 10 - 31 kg ) ( 3 × 10 5 m · s - 1 ) = 2 , 43 × 10 - 9 m

    This is the size of an atom. For this reason, electrons moving at high velocities can be used to “probe" the structure of atoms. This is discussed in more detail at the end of this chapter. [link] compares the wavelengths of fast moving electrons to the wavelengths of visible light.

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Since the de Broglie wavelength of a particle is inversely proportional to its velocity, the wavelength decreases as the velocity increases. This is confirmed in the last two examples with the electrons. De Broglie's hypothesis was confirmed by Davisson and Germer in 1927 when they observed a beam of electrons being diffracted off a nickel surface. The diffraction means that the moving electrons have a wave nature. They were also able to determine the wavelength of the electrons from the diffraction. To measure a wavelength one needs two or more diffracting centres such as pinholes, slits or atoms. For diffraction to occur the centres must be separated by a distance about the same size as the wavelength. Theoretically, all objects, not just sub-atomic particles, exhibit wave properties according to the de Broglie hypothesis.

The wavelengths of the fast electrons are much smaller than that of visible light.

Questions & Answers

what does organic mean
ntuthuko Reply
it means "from living thing"
Maqaqatu
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Refiloe Reply
It's a collision that results to a reaction
Tshepo
what is the factor and reaction rate of 20%HCL and 5%HCL
Kleng Reply
concentration and 0.05M/s
Tokozan
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Edelita Reply
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Edelita
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Malwandla Reply
forces of non-concervative
Marlet Reply
What is the following process is likely to involve carbon burning?
Sofie Reply
Combustion
Luyanda
Which Statement is part of the cell theory?
Hananiya Reply
What is the second formal statement in the cell theory
Hananiya
🤔
Bontle
what is meant by ohmic
nthabiseng Reply
when someone is referred to as ohmic, it basically means that it obeys ohm's law
Tanaka
*something
Tanaka
what is photosynthesis
Sihle Reply
is process whereby green plants to convert light energy into chemical energy.
Marcelo
what is an isomer?
Kiba Reply
organic compounds with the same molecular formula but different structural formula
liphelo
how is frequency directly proportional to kinetic energy max. and what is the relationship between the wavelength and the energy of a photon. eg if a photon with a greater wavelength and a photon with less frequency. what would be the energies. as in which one would be greater and why
Pearl Reply
relationship wave depend n enrgy greadt reflect of energy n the form of tex shape
thulare
what is Cell capacity?
liphelo Reply
what are hydrocarbons
Nyiko Reply
compounds that are made up of hydrogen and carbon only
Patience
They are compounds that consist only with hydrogen &carbon elements.
Duncan
how do i calculate the work done on an object by a frictional force
Emmelda Reply
@ Emmelda it's W = FxCos180 Where 180° is the angle between Frictional force and where the object is going, it's always 180° because it opposes the motion. Then F is the magnitude of the force and x is the displacement
Vukona

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Source:  OpenStax, Siyavula textbooks: grade 12 physical science. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11244/1.2
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