Matter Waves

Wave properties of particles
NATURE LOVES SYMMETRY

DUALITY
ENERGY
MATTER
WAVE PARTICLE PARTICLE

?
De Broglie waves or Matter waves
Louis de Broglie, ( 15 August 1892 – 19 March 1987)
was a French physicist who made groundbreaking contributions
to quantum theory.
In his 1924 PhD thesis, he postulated the wave nature
of electrons and
suggested that all matter has wave properties. When a
particle is moving, we can associate a wave with it.
This concept is known as the de Broglie hypothesis, an example of
wave–particle duality, and forms a central part of the theory of quantum
mechanics.
De Broglie won the Nobel Prize for Physics in 1929, after the wave-like
behaviour of matter was first experimentally demonstrated in 1927.
The physical explanation for the first Bohr quantization
condition comes naturally when we assume that an
electron in a hydrogen atom behaves not like a particle
but like a wave.
standing-wave condition,
L =nλ/2
If an electron in the nth Bohr orbit moves
as a wave,
2πrn=nλ/2
p=h/λ=nh/(2πrn)
=nℏ/rn.
Ln=rnp=rnnℏ /rn=nℏ
This equation is the first of Bohr’s quantization conditions. Providing a
physical explanation for Bohr’s quantization condition is a convincing
theoretical argument for the existence of matter waves.
Find the de Broglie wavelength of an electron in the ground state of

hydrogen.
When n=1 and rn=a0 =0.529Å,
λ=2πa0=2π(0.529Å)=3.324Å
λ = where = 1
√ 1− 𝑣 2 /𝑐 2
When electrons are accelerated by a voltage V
Kinetic energy ½ m v 2 = eV
De Broglie wavelength h/p=h/ 2meV
=
12.27/ V A
When accelerating voltage is 54V,
0
deBroglie wavelength of electron is 1.66 A0

Calculate the de Broglie wavelength of:
(1) a 0.65-kg basketball thrown at a speed of 10 m/s,
Nonrelativistic expression p=mv=6.5 kg/s
λ = 1.02 x10-34m/s
(2) a nonrelativistic electron with a kinetic energy of 100 eV.
Λ = 1.2 A0
(3) a relativistic electron with a kinetic energy of 108 keV.
λ=hc/ KE (KE+ 2m0c2) = 3.55pm
Rest energy of electron =m0c2
=0.511 MeV
=511keV
DAVISSON GERMER EXPERIMENT
Voltage=54V
Lattice spacing of target a=2.15 A0

deBroglie wavelength of electron is 1.66A0
DAVISSON GERMER EXPERIMENT
The main parts of the experimental setup are as follows:

•Electron gun: An electron gun is a Tungsten filament that emits electrons
via thermionic emission i.e. it emits electrons when heated to a particular
temperature.
•Electrostatic particle accelerator: Two opposite charged plates (positive
and negative plate) are used to accelerate the electrons at a known
potential.
•Collimator: The accelerator is enclosed within a cylinder that has a
narrow passage for the electrons along its axis. Its function is to render a
narrow and straight (collimated) beam of electrons ready for acceleration.
•Target: The target is a Nickel crystal. The electron beam is fired normally
on the Nickel crystal. The crystal is placed such that it can be rotated
about a fixed axis.
•Detector: A detector is used to capture the scattered electrons from the Ni
crystal. The detector can be moved in a semicircular arc as shown in the
diagram above.
and the minima of a diffraction pattern produced by X-rays. It is studied from various angles of scattering and pot
Observations of the Davisson and Germer Experiment
The detector used here can only detect the presence of an electron in the
form of a particle. As a result, the detector receives the electrons in the
form of an electronic current. The intensity (strength) of this electronic
current received by the detector and the scattering angle is studied. We call
this current as the electron intensity.
The intensity of the scattered electrons is not continuous. I

t shows a maximum and a minimum value corresponding to the maxima and the minima of
a diffraction pattern produced by X-rays. It is studied from various angles of scattering and
potential difference. For a particular voltage (54V, say) the maximum scattering happens at
a fixed angle only ( 500 ) as shown below:
Lattice spacing of target a=2.15 A0

deBroglie wavelength of electron is 1.66A0
Unlike X-ray crystallography in which X-rays penetrate the sample, in the original
Davisson–Germer experiment, only the surface atoms interact with the incident
electron beam. For the surface diffraction, the maximum intensity of the reflected
electron beam is observed for scattering angles that satisfy the condition
nλ = a sin φ
When φ is 50 and n=1, λ=1.65 A0
nλ = 2 d sin θ
What is waving in matter waves?
Probability
Of what?
Of finding the particle in a particular point at a particular time

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Wave properties of particles

NATURE LOVES SYMMETRY

WAVE PARTICLE PARTICLE

Find the de Broglie wavelength of an electron in the ground state of

When n=1 and rn=a0 =0.529Å,

De Broglie wavelength h/p=h/ 2meV

deBroglie wavelength of electron is 1.66 A0

Nonrelativistic expression p=mv=6.5 kg/s

(2) a nonrelativistic electron with a kinetic energy of 100 eV.

Lattice spacing of target a=2.15 A0

When accelerating voltage is 54V,

The main parts of the experimental setup are as follows:

Observations of the Davisson and Germer Experiment

The intensity of the scattered electrons is not continuous. I

Lattice spacing of target a=2.15 A0

When accelerating voltage is 54V,

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