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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

hi please help me how to balance redox reactions?
Brian Reply
Which equation do u wanna balance
Rifumo
i tried to write it but my phone just can't write it Cr2 O72-(aq) + H2S (aq) Cr3 (aq) + S(s)
Brian
show is the equation
Pride
us
Pride
hello. what is the difference between a primary, secondary and tertiary alcohol
Thokozani Reply
primary the C=bonded to 1 Carbon atom.... secondary=bonded to 2 carbon atoms tertiary=bonded to 3 carbon atoms
Christina
I hope you're answered
Christina
thanks Christina
Baningi
My pleasure
Christina
Thank you Christina. This is very helpful 😀👍👍💯
Thokozani
💯💯
Thokozani
Anytime Thokozani
Christina
How to calculate pH?
Thabo
thank you guys i didn't know about the primary, secondary and tertiary alcohols
Brian
thank you Christina
Brian
Hey guys which topic does AC nd DC generator falls
Phumelele
electrodynamics
Christina
thank you
Phumelele
can somebody help me with functional isomers
Baningi Reply
function isomer has the same molecular formula but different functional group
Thokozani
what's momentum
Luvelo Reply
the product of an objects mass times velocity. it is mainly prevalent in collisions.
tyrique
momentum
Sesethu
in order to produce an interference pattern, the waves used must be what?
Methane contains C and H. This compound is
Juan Reply
ketones's functional group
Moloi Reply
Why does the carboxyl group have acidc properties?
Evi Reply
carboxylic acids
tyrique
Besides improving appearance what is another reason why medal is electro plated
Elizabeth Reply
it is made of metal
Reunity
thanks
Elizabeth
what is different between Dependent variable and independent variables
Elizabeth
difference between A.c and D.c
Gerrard
AC has slip rings DC has split rings commutator
Mohludi
to avoid rust and to improve strength and conductivity
Shubham
the variable whose value doesn't depend on other is known as independent variable ex time
Shubham
are UV rays dangerous?
Khathutshelo Reply
no
Lianda
what's this?
Lianda
if there is extensive exposure, it can lead to skin cancer
tyrique
yes depend upon their intensity
Shubham
i hv a problem with this chapter because i also have problem with Newton's laws of motion
Kearabetswe Reply
I have a problem with calculation in Newton's law of motion
Vuyokazi
me also VuyokAzi
Amahle
what's the question
Christina
Which area in particular?
Kevo
just get the hang if free body diagrams,know the directions.
Muregu
I know how to draw free diagram I have a problem with calculation in second law
Vuyokazi
what is the wave model of atom ?
Trazy Reply
guys what is momentum
Luvelo
@Luvelo... Momentum is the product of an object's mass and its velocity.
Christina
the model based on wave nature of electron
Shubham
Can you state the Doppler effect in words
Nombuyiselo Reply
State Coulomb's law in words
Nombuyiselo
the magnitude of the electrostatic force exerted point on each other is directly proportional to the product on magnitude of charges and inversely proportional to the square distance between them
Cassiey
the change in frequency of the sound detected by a listener because sound source and listener have different velocity
Elizabeth
Newton's first law of motion
Nnuso Reply
When an object is at rest or traveling at a constant velocity it will remain at rest unless an unbalanced force acts upon it.
Tumelo
yes that's true
Mphoko
Always know Newton's second law of motion. It appears to be in every question paper
Botho
Is the normal force always 0
Mpilo Reply
no. newtons 3rd law states that if something exerts a force into something else it will experience that same magnitude of force but in the opposite direction. so the net force is equal to the force the object applies to the surface but in the opposite direction
Courtney
define the term rate of reaction in word
David Reply
I think it a chemical process in which substance act mutually on each other.
Sphesihle
how many bones are in the human body?
Asali
it's the rate at which the reactants are able to change to products
Courtney
206 bones
Elizabeth

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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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