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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

how is ester formed
Aubrey Reply
how is n ester formed
Aubrey
Alcohol reacts with a carboxylic acid
Texas
an athlete with a mass of 70kg runs at a velocity of 45km . determine the athlete's momentum
Lesedi Reply
Is that a velocity or something else
msawenkosi
45km/h i guess
Texas
Change to m/s
Texas
45km/h = 12.5 m/s p=mv =70×12.5 =875 kg.m/s
Thato
what are the measures of the rates of reaction
Lesego Reply
Volume Concentration Temperature Pressure Surface Area
Thato
the principle of superposition of waves
Sfundo Reply
what is work
Kool Reply
is this a group chat
Nobuhle Reply
Hey can y'all define newton's 2nd law
mthebzification
If a resultant force act on an object...the object will accelerate in the direction of a resultant force,the acceleration of the object is directly proportional to the net force and inversely proportional to the mass of the object
mosa
how do you calculate tension force
Bulumko
use the formula Fnet=ma if there is tension connecting two objects
Sboniso
to calculate Tension, usually calculate acceleration first Draw separate free body diagrams for each body. Apply Fnet = ma to calculate Tension
Kevin
Hi people
Paul
how does temperature affect the equilibrium position
Blessing Reply
an increased temperature increases the average kinetic energy thus in turn increases the number of effective collisions........
Lwando
so...which reaction is favored between endothermic and exothermic .when temperature is increased..?
Blessing
exothermic reaction because energy is realised to the surroundings as heat and light energy ....graphical so much energy is realised as reactants to form product and because temperature is high rate of reaction is fast which means there is a successful collision
Code
INTEMENDO - INCREASE IN TEMPERATURE FAVOURS ENDOTHERMIC DETEMEXO - DECREASE IN TEMPERATURE FAVOURS EXOTHERMIC
Thato
an object will continue in a state of rest unless it is acted upon an unbalanced force
Junior Reply
Newton's Law 1
Code
First Newton's Law
Azola
Newton's first law
Surprise
newton first law
Thinavhuyo
Newton's first law
Blessing
when pressure is increased what happen to volume
Siphelo Reply
decreases
Code
care to explain?
Mpati
if pressure is applied to a pistol , the volume will decrease and particles will collide more frequently with the wall of a container .Each time they collide with the wall they exert a force on them .More collision means more force and the pressure will increase , that Boyle's Law
Code
Because the volume has decreased , the particle will collide more frequently with the wall of a container and each time they collide with the wall of a container they exert a force on them.More collision means more force so the pressure will increase , that Boyle's Law
Code
what is the difference between momentum and a change in momentum?
Chavonne Reply
How to name a branched molecule from right or left?
Vadin Reply
What's Free Fall
Senzo Reply
Free Fall means there is no acting force on that object.
Dingaletu
only gravitational force
Dingaletu
no external force acting on an object
Sphiwe
by only force of gravite
Sello
but gravitational force
Sphiwe
true
Lucky
a motion in which the only force acting is gravitational force
Blessing
and an object experiencing free fall is referred as a projectile
Blessing
Do polymers form restrictedly only if compound is saturated, only?
milani Reply
what is a free fall?
Beyanca Reply
is when The Only Force acting On an Object is Gravitational Force
Madman
Thats right
Beyanca
then Why ask when you Know the answer?
Madman
She's just helping those who forgot it...bro
Thato
guys I need help on Getting ready for a last minute test
Kenelioe Reply
what help you need
Neil
we'll I'm in grade 12 so we doing this topic about upac thing
Kenelioe
on What?
Madman
the organic molecule section
Kenelioe
IUPAC NAMING WHICH FUNCTIONAL GROUP YOU CANNOT NAME?SO I COULD HELP YOU
Madman
ester
Sboniso
you should also look at structural isomers. Its crucial that they might add that one. also try and write down the structural formula of all the given compounds on the table
milani

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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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