Chapter 27 Early Quantum Theory and Models of the Atom

Chapter 27 

Early Quantum Theory and 

Models of the Atom


• Discovery and Properties of the Electron 

• Planck’s Quantum Hypothesis and 

Blackbody Radiation 

• Photon Theory of Light and 

Photoelectric Effect 

• Energy, Mass, and Momentum of a 

Photon 

• Compton Effect 

• Photon Interactions; Pair Production


• WaveParticle Duality; 

the Principle of Complementarity 

• Wave Nature of Matter 

• Electron Microscopes 

• Early Models of the Atom 

• Atomic Spectra: 

Key to the Structure of the Atom 

• The Bohr Model 

• de Broglie’s Hypothesis Applied to Atoms

Discovery and Properties of the Electron 

“Cathode rays” were discovered in late 1800’s and 

found to be deflected by electric or magnetic fields in 

ways that implied particles with a negative charge.


By accelerating the rays through a known potential 

and then measuring the radius of their path in a 

known magnetic field, the charge to mass ratio was 

measured: 

(271) 

The result is


The currently accepted value of e is: 

Knowing e and the e/m ratio allows the electron’s 

mass to be calculated. Result is

Planck’s Quantum Hypothesis and 


All objects emit radiation whose total intensity is 

proportional to the fourth power of their temperature. 

This is called thermal radiation; a “blackbody” emits 

only thermal radiation. 

The spectrum of 

blackbody radiation 

is in this figure for a 

few temperatures. 

The frequency of 

peak intensity 

increases linearly 

with temperature.



This spectrum could not be reproduced using the 

physics known in the late 19 th century. 

A solution was proposed by Max Planck in 1900: 

The energy of oscillations within atoms cannot have 

an arbitrary value; it is related to the frequency: 

The constant h is now called Planck’s constant.



Planck found the value of this constant by fitting 

blackbody curves for a number of temperatures 

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.

Photon Theory of Light and the 


Einstein suggested that ,given the success of Planck’s 

theory, light must be emitted in small energy packets: 

These tiny packets of energy are like a particle 

and are called photons. 

So is light a wave or a particle



Evidence for Particle 

The photoelectric 

effect: When light strikes 

a metal, electrons are 

emitted. But, this effect 

does not occur if the 

frequency of the light is 

too low. The kinetic 

energy of an electron 

increases with frequency.



If light is a wave, theory predicts: 

• Number of electrons and their energy should 

increase with intensity 

• Frequency would not matter 

• Also one can get Interference and Diffraction 

from waves but not from particles



If light is made up of particles, theory predicts: 

• Increasing intensity increases number of 

electrons but not energy 

• Above a minimum energy required to break 

atomic bond, kinetic energy will increase linearly 

with frequency 

• There is a cutoff frequency below which no 

electrons will be emitted, regardless of intensity



The particle theory assumes that an electron 

absorbs a single photon with energy hf. 

Plotting the kinetic energy vs. frequency: 

This shows clear 

agreement with the 

photon theory, and 

not with wave 

theory.

Energy, Mass, and Momentum of a Photon 

Clearly, a photon must travel at the speed of light. 

Looking at the relativistic equation for momentum, 

From relativity it is clear that this can only happen 

if a photon’s rest mass is zero 

We already know that the energy is hf; we can put 

this in the relativistic energy momentum relation 

and find the momentum:

The Compton Effect 

Compton scattered Xrays from different materials. 

He found that the scattered Xrays had a slightly 

longer wavelength than the incident ones, and that 

the wavelength depended on the scattering angle:

The Compton Effect 

This is another effect that is correctly predicted by 

the photon model and not by the wave model.

Photon Interactions and Pair Production 

Photons passing through matter can undergo the 

following interactions: 

• Photoelectric effect: photon is completely 

absorbed, electron is ejected 

• Photon may be totally absorbed by electron, but 

not have enough energy to eject it; the electron 

moves into an excited state 

• The photon can scatter from an atom and lose 

some energy 

• The photon can produce an electronpositron pair.

Photon Interactions and Pair Production 

In pair production, energy, electric charge, and 

momentum must all be conserved. 

Energy will be conserved 

through the mass and kinetic 

energy of the electron and 

positron; their opposite charges 

conserve charge; and the 

interaction must take place in 

the electromagnetic field of a 

nucleus, which can contribute 

momentum.

WaveParticle Duality 

The Principle of Complementarity 

We have phenomena such as diffraction and 

interference that show that light is a wave, and 

phenomena such as the photoelectric effect and the 

Compton effect that show that it is a particle. 

How do we decide which one it actually is 

The question has no answer one way or the other ! 

We must accept the dual waveparticle nature of light 

based on the experimental results. 

So, are what we think of as particles also waves

WaveParticle Duality 

The Principle of Complementarity 

The principle of complementarity states that both 

the wave and particle aspects of light are 

fundamental to its nature. 

Whether we “see” waves or particles is just our 

interpretation of how light behaves in various types 

of measurements – based on an incomplete theory.

Wave Nature of Matter 

Just as light sometimes behaves as a particle, 

matter sometimes behaves like a wave. 

The wavelength of a particle of matter is also 

This wavelength is extraordinarily small for any 

object that has any significant mass, but 

becomes more important for very light particles 

such as an electron. Electrons show the same 

diffraction from crystals as found for xrays.

Electron Microscopes 

The wavelength of electrons will vary with energy, 

but is still quite short. This makes electrons useful 

for imaging – remember that the smallest object 

that can be resolved is about one wavelength. 

Electrons used in electron microscopes have 

wavelengths of about 0.004 nm. 

This means it is possible to “see” an individual 

atom with some electron based techniques


Transmission 

electron microscope 

the electrons are first 

accelerated by a 

potential difference 

before encountering 

the object, 

and are focused by 

a magnetic “lens” 

to form the image.


Scanning electron 

microscope – 

the electron beam is 

scanned back and 

forth across the 

object to be imaged


Scanning tunneling microscope – up and down 

motion of the probe keeps the current constant. 

Plotting that motion 

produces an image 

of the surface.

Early Models of the Atom 

It was known that atoms were electrically neutral, but 

that they could become charged, implying that there 

were positive and negative charges and that some of 

them could be removed. 

A popular atomic model was 

this “plumpudding” model:


Rutherford did an experiment now known as 

Rutherford scattering that showed that the 

positively charged nucleus must be extremely 

small compared to the rest of the atom. 

He scattered alpha particles – helium nuclei – from 

a metal foil and observed the scattering angle. He 

found that some of the angles were far larger than 

the plumpudding model would allow. The 

expanation was that the positively charged region 

had to be tiny compared to the atoms overall size.

Rutherford Scattering Experiment 

The only way to account for the large angles was to 

assume that all the positive charge was contained 

within a tiny volume – now we know that the radius 

of the nucleus 

is 1/10000 that 

of the atom.


Therefore, Rutherford’s 

model of the atom is 

mostly empty space: 

Problem: accelerated 

electron should radiate 

all of the time if moving 

in circular motion.

Atomic Spectra: Key to Atom’s Structure 

A very thin gas heated in a discharge tube emits 

light only at characteristic frequencies.

Atomic Spectra Provided the Key to the 

Structure of the Atom 

An atomic spectrum is a line spectrum – only 

certain frequencies appear. If white light passes 

through such a gas, it absorbs at those same 

frequencies.


The wavelengths of electrons emitted from hydrogen 

have a regular pattern: 

This is called the Balmer series. R is the Rydberg 

constant:


Other series include the Lyman series: 

and the Paschen series:


A portion of the complete spectrum of hydrogen is 

shown here. The lines cannot be explained by the 

Rutherford theory without some extra constraints.

The Bohr Model of an Atom 

Bohr proposed that the possible energy states for 

atomic electrons were quantized – only certain 

values were possible. Then the spectrum could be 

explained as transitions from one level to another.


Bohr also found that the angular momentum 

was quantized in an orbital picture of the atom:


An electron is held in orbit by the Coulomb force:

The Bohr Atom 

Using the Coulomb force, one can easily calculate 

the radii of the orbits:

The Bohr Model of a Hydrogen Atom 

The lowest energy level is 

called the ground state; 

the others are excited 

states.

de Broglie’s Hypothesis Applied to Atoms 

De Broglie’s hypothesis is the one associating a 

wavelength with the momentum of a particle. 

He proposed that only those orbits where the wave 

would be a circular standing wave will occur. This 

yields the same relation that Bohr had proposed. 

These ideas are the start of current understanding of 

the electronic structure of an atom but did not yet 

incorporate the idea of electronic wavefunctions. 

They did have the energy & angular momentum 

properly quantized – that’s why they work so well.

de Broglie’s Hypothesis Applied to 

Atoms in Bohr’s Model 

These are circular 

standing waves for 

n = 2, 3, and 5. 

Explains why electrons 

do not radiate, as one 

expects from an 

accelerating charge. 

Emission occurs only 

for a state change, ∆n

Summary of Chapter 27 

• Planck’s hypothesis to explain blackbody spectrum: 

molecular oscillation energies are quantized 

• Light can be considered to consist of photons, 

each of energy 

• Photoelectric effect: incident photons knock 

electrons out of material when incident light 

is above some threshold frequency


• Compton effect and pair production also support 

photon theory 

• Waveparticle duality – both light and matter have 

both wave and particle properties 

• The de Broglie wavelength of an object:


• Principle of complementarity: both wave and 

particle properties are necessary for complete 

understanding 

• Rutherford showed that atom has tiny nucleus 

• Line spectra are explained by electrons having 

only certain specific orbits L is quantized 

• Ground state has the lowest energy; the others are 

called excited states

Chapter 27 Early Quantum Theory and Models of the Atom

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