ChaPtER 2 ChaPtER 2 ::

EARLY QUANTUM EARLY QUANTUM

THEORYTHEORY

SCOPE OF STUDYSCOPE OF STUDY 7 sub- topics students should learn and understand :

Electrons

J.J. Thompson’s experiment, Milikan experiment

De Broglie Relation

Wave-particle duality; the Principle of Complementarily

Wave nature of matter

Electron diffraction

Planck’s quantum hypothesis

introductionintroductionQuantum Physics:

developed early 20th century, in response to shortcomings of

classical physics in describing certain phenomena (blackbody

radiation, photoelectric effect, emission and absorption spectra…)

describes “small” objects (e.g. atoms and their constituents)

QP is “weird and counterintuitive”

“Those who are not shocked when they first come across quantum

theory cannot possibly have understood it” (Niels Bohr)

“Nobody feels perfectly comfortable with it “ (Murray Gell-Mann)

“I can safely say that nobody understands quantum mechanics”

(Richard Feynman)

introductionintroductionQuantum physics is basically the recognition that there is less difference

between waves and particles than was thought before.

Key insights:

• light can behave like a particle

• particles (e.g. electrons) are indistinguishable

• particles can behave like waves (or wave packets)

• waves gain or lose energy only in "quantized amounts“

• detection (measurement) of a particle wave will change

suddenly into a new wave

• quantum mechanical interference – amplitudes add

• QP is intrinsically probabilistic

• what you can measure is what you can know

electronselectrons

PROPERTIESPROPERTIESPROPERTIESPROPERTIES

Tiny and very

light particles

Negative electric

charge

Mass : 9.11 ×

10 −31 kilograms.

Has spin creates

magnetic field Wave

Particles

electronselectrons

Electrons carry one unit of negative elementary charge.

Since an electric or magnetic field exerts a force on electrons, it a ects their ff

motion.

In spite of the fact that electric fields are everywhere around us, in every day

life we do not directly observe the motion of electrons.

However, such observations are possible with a device called a cathode-ray

tube (CRT), which is just a technical name for a tube present in most of TV sets

or computer monitors.

J.J.THOMSON’S EXPERIMENT J.J.THOMSON’S EXPERIMENT

J.J. Thomson (1856-1940)J.J. Thomson (1856-1940)

J.J.THOMSON’S EXPERIMENT J.J.THOMSON’S EXPERIMENT

J.J. Thomson was born on December 18, 1856 in Manchester.

The major accomplishment of J.J. Thomson was the discovery of the electron.

An electron is a particle that is smaller than an atom and is negatively charged.

Thomson created the plum pudding model, which was the way he thought the

structure of the atom was made up.

The tiny negatively charged atoms were embedded in a positively charged

cloud.

(J.J. Thomsons Atom Model) (J.J. Thomsons Atom Model) 

MILIKAN EXPERIMENT MILIKAN EXPERIMENT

Robert Andrews Millikan (1868-1953)Robert Andrews Millikan (1868-1953)

Millikan made numerous momentous discoveries.

Mainly in the fields of electricity, optics, and molecular physics.

A major success was the accurate determination of the charge carried by an

electron, using the elegant "falling-drop method".

He also proved that this quantity was a constant for all electrons (1910), thus

demonstrating the atomic structure of electricity.

He verified experimentally Einstein's all-important photoelectric equation, and

made the first direct photoelectric determination of Planck's constant.  

MILIKAN EXPERIMENT MILIKAN EXPERIMENT

MILIKAN EXPERIMENT MILIKAN EXPERIMENT

Droplets of oil could electrify themselves owing to friction when they were

atomised.

They could also get the charge by X-raying.

Millikan put some electric potential difference on plates generating electric field

between them.

DE BROGLIE RELATION DE BROGLIE RELATION

A particle representing a quantum of light or other

electromagnetic radiation  regarded as a particle with

zero rest mass and charge, unit spin, and energy equal

to the product of the frequency of the radiation and

the Planck constant.

PHOTONPHOTON

DE BROGLIE RELATION DE BROGLIE RELATION

Prince Louis de Broglie (1892 – 1987)Prince Louis de Broglie (1892 – 1987)

DE BROGLIE RELATION DE BROGLIE RELATION

 In 1923, Frenchman Louis de Broglie proposed the particles of matter also had

wavelengths and could behave as waves, just as photons did.

He stated that under special relativity :

The photon, a particle of energy, had a wavelength associated with it.

The electron, particles of matter, such as electrons also had a wavelength.

DE BROGLIE RELATION DE BROGLIE RELATION

λ = h/p = h/mvλ = h/p = h/mv

de Broglie wavelength

Planck’s constant6.63 x 10-34 J/s

mass

velocitywavelenght

momentum

DE BROGLIE RELATION DE BROGLIE RELATION

Example: The de Broglie Wavelength of an Electron and a Baseball

Determine the de Broglie wavelength of (a) an electron moving at a speed of

6.0x106 m/s and (b) a baseball (mass = 0.15 kg) moving at a speed of 13 m/s.

m102.1

sm100.6kg101.9

sJ1063.6 10631

34

ph

m103.3

sm13kg 15.0

sJ1063.6 3434

ph

Solution :

Wave particle duality Wave particle duality

In 1927, Neils Bohr formulated his "principle of complementarity*' which

brought together the wave like properties of matter and the particle like

properties of light into a coherent theoretical framework often called

"wave-particle duality" or the "Copenhagen interpretation."

In 1927, Neils Bohr formulated his "principle of complementarity*' which

brought together the wave like properties of matter and the particle like

properties of light into a coherent theoretical framework often called

"wave-particle duality" or the "Copenhagen interpretation."

HISTORY

Wave particle duality Wave particle duality

Niels Bohr (1885-1962) Niels Bohr (1885-1962)

Wave particle duality Wave particle duality

His starting point was the impossibility to distinguish satisfactorily

between the actual behavior of atomic objects, and their interaction with the

measuring instruments that serve to define the conditions under which the

phenomena appear.

Examine light with one instrument, the argument went, and it undulates

like a wave; select another and it scatters like a particle.

His conclusion was that evidence obtained under different experimental

conditions cannot be comprehended within a single picture, but must be

regarded as complementary in the sense that only the totality of the

phenomenon exhausts the possible information about the objects.

Wave particle duality Wave particle duality

Wave Particle DualityWave Particle Duality

Principle of ComplementarityPrinciple of Complementarity

A system can exhibit wave-like behavior and particle-

like behavior, but no experiment could demonstrate both

behaviors simultaneously.

A system can exhibit wave-like behavior and particle-

like behavior, but no experiment could demonstrate both

behaviors simultaneously.

Wave NATURE OF Wave NATURE OF MATTER MATTER

Remember that :

Matter has dual character

With every body there is connected some wave. 

The lower mass and the lower velocity the body has, the

longer is its wave.

Louis De Broglie created the theory of the dual nature of matter.

Remember that :

Matter has dual character

With every body there is connected some wave. 

The lower mass and the lower velocity the body has, the

longer is its wave.

Louis De Broglie created the theory of the dual nature of matter.

Wave NATURE OF Wave NATURE OF MATTER MATTER

Just as light sometimes behaves like a particle, matter sometimes

behaves like a wave.

The wavelength of a particle of matter is

.

This wavelength is extraordinarily small.

Wave NATURE OF Wave NATURE OF MATTER MATTER

Example: Wavelength of a ball.

Calculate the de Broglie wavelength of a 0.20-kg ball moving with a speed

of 15 m/s.

Solution:

λ = h/p = 2.2 x 10-34 m.

Wave NATURE OF Wave NATURE OF MATTER MATTER

Example : Wavelength of an electron.

Determine the wavelength of an electron that has been accelerated

through a potential difference of 100 V.

Solution:

The kinetic energy of the electron is classical. Its speed is found

from conservation of energy: ½ mv2 = eV, so v = 5.9 x 106 m/s.

The wavelength is then h/p = 1.2 x 10-10 m; roughly the size of an

atom.

PLANCK’S QUANTUM PLANCK’S QUANTUM HYPOTHESIS HYPOTHESIS

Max Karl Ernst Ludwig Planck (1858-1947)Max Karl Ernst Ludwig Planck (1858-1947)

PLANCK’S QUANTUM PLANCK’S QUANTUM HYPOTHESIS HYPOTHESIS

He suggested that the energy of atomic oscillations within atoms cannot

have an arbitrary value; it is related to the frequency:

The constant h is now called Planck’s constant and n is called a

quantum number (discrete number)

The constant h is now called Planck’s constant and n is called a

quantum number (discrete number)

PLANCK’S QUANTUM PLANCK’S QUANTUM HYPOTHESIS HYPOTHESIS

Planck found the value of his constant by fitting blackbody curves

to the formula

where : I (λ , T) = radiation intensity as a function of wavelength at the

temperature, T

k = Boltzman’s constant

c = speed of light

h = Planck’s constant

PLANCK’S QUANTUM PLANCK’S QUANTUM HYPOTHESIS HYPOTHESIS

Planck’s proposal was that the energy of an oscillation had to be an

integral multiple of hf. This is called the quantization of energy.

The quantum hypothesis states that the energy of an oscillator can be E

= hf, or 2hf, or 3hf and so on but there cannot be vibrations with

energies between these values.

Energy would not be a continuous quantity.

Electron diffraction Electron diffraction

A collective scattering phenomenon with

electrons being (nearly elastically)

scattered by atoms in a regular array

(crystal).

A collective scattering phenomenon with

electrons being (nearly elastically)

scattered by atoms in a regular array

(crystal).

DEFINITION DEFINITION

Electron diffraction Electron diffraction

This can be understood in analogy to the Huygens principle for the

diffraction of light.

The incoming plane electron wave interacts with the atoms, and

secondary waves are generated which interfere with each other.

This occurs either constructively (reinforcement at certain

scattering angles generating diffracted beams) or destructively

(extinguishing of beams). 

As in X-ray diffraction (XRD), the scattering event can be

described as a reflection of the beams at planes of atoms (lattice

planes). 

Electron diffraction Electron diffraction

The Bragg law gives the relation between interplanar distance d and

diffraction angle θ:

The Bragg law gives the relation between interplanar distance d and

diffraction angle θ:

n λ = 2 d sin θn λ = 2 d sin θ

Electron diffraction Electron diffraction Example :

Assume that the electrons strike perpendicular to the surface of a solid,

and that their energy is low, K = 100 eV, so that they interact only with

the surface layer of atoms. If the smallest angle at which a diffraction

maximum occurs is at 24°, what is the separation d between the atoms on

the surface?

Solution:

The smallest angle will occur when d sin θ = λ. The electrons are not

relativistic, so the wavelength can be found from the kinetic energy: λ =

0.123 nm. Then the spacing is λ/sin θ = 0.30 nm.

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:: PrOBLeMS onLY eXisT iN tHe hUmaN MinD ::~ ~ Anthony de Mello ~ ~