1. Limitations of Classical Physics

2. Bohr's Atomic Model

Bohr's Atomic Model

In 1913, Danish physicist Niels Bohr proposed a new model of the atom that combined Rutherford's nuclear model with the emerging idea of quantized energy.

Rutherford had shown that atoms contain a tiny, positively charged nucleus surrounded by electrons. However, classical physics could not explain why those electrons remained stable or why atoms produced distinctive line spectra.

Bohr proposed a major new idea:

Electrons can occupy only certain allowed energy states.

These discrete states are called:

quantized energy levels

---

From Rutherford to Bohr

Rutherford's model described the atom as:

• a tiny positive nucleus
• electrons surrounding the nucleus
• mostly empty space

This successfully explained alpha-particle scattering.

However, Rutherford's model had a serious problem.

According to classical electromagnetism, an electron moving in a circular path is accelerating.

An accelerating charged particle should emit electromagnetic radiation.

Therefore, classical physics predicted:

electron radiates energy → electron loses energy → orbit shrinks → electron falls into nucleus

Real atoms do not behave this way.

Bohr's model was designed partly to solve this:

atomic stability problem

---

Bohr's Main Ideas

Bohr proposed several important ideas.

Electrons occupy allowed states

An electron cannot have just any energy.

It can occupy only certain:

discrete energy levels

Stable states do not continuously radiate

An electron remaining in an allowed state does not continuously lose energy as predicted by the classical model.

Electrons can change energy levels

An electron can move from one allowed state to another.

Energy is exchanged during transitions

When an electron changes energy level, electromagnetic radiation can be:

• absorbed
• emitted

The photon energy equals the difference between the two atomic energy levels.

---

Quantized Orbits

In the original Bohr model, electrons were pictured moving in specific allowed circular orbits around the nucleus.

Each orbit corresponded to a particular:

energy

The allowed states were labelled using the principal quantum number:

n = 1, 2, 3, 4, ...

The lowest energy state is:

n = 1

This is called the:

ground state

Higher values of n represent higher-energy states.

---

Energy Levels Are Discrete

Imagine a staircase.

You can stand on:

• step 1
• step 2
• step 3

but not halfway between two steps.

Bohr suggested that atomic energy behaves similarly.

An electron can occupy:

E₁, E₂, E₃, E₄ ...

but cannot have an arbitrary energy between allowed levels.

This is what is meant by:

quantization

---

Energy Levels Are Not Equally Spaced

The allowed energy levels become closer together as n increases.

For hydrogen:

Eₙ = −13.6/n² eV

where:

• Eₙ = energy of level n
• n = principal quantum number
• eV = electron volt

For example:

Notice that the energy levels become closer together as n increases.

---

Why the Energies Are Negative

The zero-energy reference represents an electron completely separated from the atom.

A bound electron has:

negative energy

Energy must be added to remove it from the atom.

At:

E = 0

the electron is no longer bound.

---

Ground State

The lowest possible energy level is:

n = 1

For hydrogen:

E₁ = −13.6 eV

This is the atom's:

ground state

The electron cannot move to a lower energy state because no lower allowed state exists.

This helps explain why the hydrogen atom is stable.

---

Excited States

If an electron absorbs energy, it can move to a higher energy level.

This produces an:

excited state

For example:

n = 1 → n = 2

or:

n = 2 → n = 4

The electron does not gradually move through every possible intermediate energy.

It changes from one allowed energy state to another.

---

Absorption

For an electron to move to a higher energy level, it must absorb exactly the required energy.

For example:

lower energy → absorb photon → higher energy

The required photon energy is:

E_photon = E_high − E_low

or:

ΔE = hf

where:

• ΔE = difference between energy levels
• h = Planck's constant
• f = photon frequency

---

Emission

An electron in an excited state can move to a lower energy level.

When it does:

higher energy → lower energy + photon

The energy of the emitted photon is:

E_photon = E_high − E_low

Therefore:

ΔE = hf

The atom cannot emit arbitrary photon energies because only certain energy differences are available.

---

Photon Energy

The energy of a photon is:

E = hf

where:

• E = photon energy
• h = 6.63 × 10⁻³⁴ J·s
• f = frequency

Because:

c = fλ

photon energy can also be written:

E = hc/λ

Therefore:

larger energy transition → higher frequency → shorter wavelength

and:

smaller energy transition → lower frequency → longer wavelength

---

Worked Example: Energy Between Levels

For hydrogen:

n = 1:

E₁ = −13.6 eV

n = 2:

E₂ = −3.40 eV

How much energy must be absorbed for the transition:

n = 1 → n = 2?

Calculate:

ΔE = E₂ − E₁

ΔE = −3.40 − (−13.6)

ΔE = 10.2 eV

Therefore, the atom must absorb a photon with energy:

10.2 eV

---

Worked Example: Emission

Suppose an electron falls from:

n = 3 → n = 2

For hydrogen:

E₃ = −1.51 eV

E₂ = −3.40 eV

Energy released:

ΔE = E₃ − E₂

ΔE = −1.51 − (−3.40)

ΔE = 1.89 eV

A photon with energy approximately:

1.89 eV

is emitted.

---

Converting Electron Volts to Joules

Atomic energies are often measured in:

electron volts (eV)

The conversion is:

1 eV = 1.60 × 10⁻¹⁹ J

For the previous transition:

E = 1.89 eV

E = 1.89 × 1.60 × 10⁻¹⁹

E ≈ 3.02 × 10⁻¹⁹ J

---

Finding the Photon Frequency

Using:

E = hf

Rearrange:

f = E/h

For:

E = 3.02 × 10⁻¹⁹ J

f = (3.02 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 4.56 × 10¹⁴ Hz

This frequency corresponds to visible light.

---

Finding the Wavelength

Use:

c = fλ

Therefore:

λ = c/f

λ = (3.00 × 10⁸)/(4.56 × 10¹⁴)

λ ≈ 6.58 × 10⁻⁷ m

or:

658 nm

This is approximately the red hydrogen spectral line associated with the n = 3 → n = 2 transition.

---

Atomic Spectra

One of Bohr's greatest successes was explaining the:

line spectrum of hydrogen

When hydrogen gas is excited, it emits light at specific wavelengths rather than producing every possible wavelength.

The visible hydrogen spectrum contains several characteristic lines.

These lines occur because hydrogen electrons can make only certain:

energy transitions

---

Why the Spectrum Is Not Continuous

Suppose atomic energy could take any value.

Electrons could then lose any amount of energy.

The emitted photons could have almost any energy.

We would expect a:

continuous spectrum

Instead, experiments show:

specific spectral lines

Bohr explained this using quantized energy levels.

---

From Energy Levels to Spectral Lines

The process is:

electron begins in higher level

↓

electron moves to lower level

↓

specific energy difference released

↓

photon emitted with E = hf

↓

specific frequency

↓

specific wavelength

↓

spectral line

Therefore:

each spectral line corresponds to a particular electron transition.

---

The Hydrogen Spectrum

Different groups of hydrogen spectral lines correspond to transitions ending at different energy levels.

Important series include:

Lyman series

Transitions ending at:

n = 1

These are mainly in the:

ultraviolet

Balmer series

Transitions ending at:

n = 2

Several are in the:

visible spectrum

Paschen series

Transitions ending at:

n = 3

These are mainly in the:

infrared

---

The Balmer Series

The visible lines of hydrogen include transitions such as:

n = 3 → n = 2

n = 4 → n = 2

n = 5 → n = 2

n = 6 → n = 2

Each transition releases a different amount of energy.

Therefore each produces a different:

wavelength

This creates the distinctive hydrogen spectrum.

---

Absorption Spectra

Atoms can also absorb photons.

If a photon has exactly the right energy to move an electron to a higher allowed state:

photon absorbed → electron moves upward

For example:

n = 2 → n = 3

requires exactly the same energy as is emitted during:

n = 3 → n = 2

Therefore atoms absorb and emit characteristic wavelengths.

---

Emission vs Absorption

Emission

Electron:

high → low

Photon:

released

Absorption

Electron:

low → high

Photon:

absorbed

In both cases:

photon energy = difference between allowed energy levels

---

Ionization

If enough energy is supplied, an electron can be completely removed from the atom.

This is:

ionization

For hydrogen in its ground state:

E₁ = −13.6 eV

The free-electron level is:

E = 0 eV

Therefore, the minimum energy needed to ionize ground-state hydrogen is:

13.6 eV

---

Worked Example: Ionization

How much energy is needed to remove the electron from ground-state hydrogen?

Initial energy:

−13.6 eV

Final energy:

0 eV

Therefore:

ΔE = 0 − (−13.6)

ΔE = 13.6 eV

This is the:

ionization energy

of hydrogen from its ground state.

---

Angular Momentum Quantization

Bohr also proposed that electron angular momentum could take only certain values.

In the Bohr model:

L = nħ

where:

• L = electron angular momentum
• n = 1, 2, 3, ...
• ħ = h/(2π)

This restriction helped determine the allowed electron orbits.

It was a major departure from classical mechanics, in which orbital properties could vary continuously.

---

Radius of the Hydrogen Atom

Bohr's model also predicts specific allowed orbital radii.

For hydrogen:

rₙ = n²a₀

where:

a₀ ≈ 5.29 × 10⁻¹¹ m

is the:

Bohr radius

For n = 1:

r₁ = 5.29 × 10⁻¹¹ m

For n = 2:

r₂ = 4a₀

For n = 3:

r₃ = 9a₀

Thus higher-energy states correspond, in the Bohr picture, to larger electron orbits.

---

Strengths of Bohr's Model

Bohr's model represented a major advance in atomic physics.

One major strength was that it explained:

atomic stability

Electrons could remain in allowed states without continuously radiating energy.

---

Explaining Hydrogen's Spectrum

Another major strength was its successful explanation of:

hydrogen's line spectrum

The model correctly connected:

energy levels → electron transitions → photon energies → spectral wavelengths

This was strong evidence that atomic energies are quantized.

---

Predicting Hydrogen Energies

Bohr's model correctly predicts the hydrogen energy levels:

Eₙ = −13.6/n² eV

This allowed physicists to calculate:

• transition energies
• photon frequencies
• photon wavelengths
• ionization energy

Its quantitative success for hydrogen was remarkable.

---

Explaining the Rydberg Formula

Before Bohr, the wavelengths of hydrogen spectral lines could be described experimentally using mathematical relationships such as the:

Rydberg formula

Bohr's model provided a physical explanation for why this pattern existed.

For hydrogen:

1/λ = R(1/n_f² − 1/n_i²)

where:

• R = Rydberg constant
• n_i = initial higher level
• n_f = final lower level

This connected experimental spectroscopy with atomic structure.

---

Hydrogen-Like Ions

Bohr's model also works reasonably well for systems containing only one electron.

Examples include:

• hydrogen, H
• singly ionized helium, He⁺
• doubly ionized lithium, Li²⁺

These are called:

hydrogen-like ions

The model becomes much less successful when several electrons interact with one another.

---

Bohr's Model Was Not the Final Model

Despite its successes, Bohr's model had important limitations.

Its picture of electrons travelling around the nucleus in precise circular paths is not the modern description of atomic structure.

Modern quantum mechanics describes electrons using:

wavefunctions and probability distributions

rather than simple planetary-style orbits.

---

Limitation: Multi-Electron Atoms

Bohr's model works best for:

hydrogen

and other one-electron systems.

It does not accurately predict the spectra of atoms containing many electrons.

In multi-electron atoms, electrons interact with:

• the nucleus
• other electrons

These interactions are much more complicated than Bohr's simple model allows.

---

Limitation: Electron-Electron Interactions

Electrons repel one another electrically.

Bohr's simple model does not adequately describe these:

electron-electron interactions

As more electrons are added to an atom, their behaviour cannot be explained simply by assigning independent circular orbits.

---

Limitation: Detailed Spectral Structure

With improved instruments, scientists discovered that some spectral lines actually consist of several closely spaced lines.

This is called:

fine structure

Bohr's original model could not fully explain these details.

---

Limitation: Magnetic Fields

When atoms are placed in magnetic fields, some spectral lines split.

This is associated with the:

Zeeman effect

The original Bohr model could not adequately explain all of these observations.

Additional quantum ideas were required.

---

Limitation: Electric Fields

Spectral lines can also split when atoms are placed in external electric fields.

This is called the:

Stark effect

Again, the simple Bohr model was insufficient.

---

Limitation: Fixed Electron Paths

Perhaps the most important conceptual limitation is the idea of a precisely defined electron orbit.

Modern quantum mechanics does not assign an electron a simple circular path around the nucleus.

Instead, it describes the probability of detecting an electron in different regions of space.

These regions are associated with:

atomic orbitals

---

Orbit vs Orbital

These terms are different.

Orbit

In the Bohr model:

a defined path followed by an electron

Orbital

In quantum mechanics:

a quantum state associated with a probability distribution for an electron

An orbital is not simply a circular track around the nucleus.

---

The Modern Electron Cloud

The modern atomic model is sometimes represented using an:

electron cloud

This does not mean that electrons are physically spread out like ordinary fog.

It represents the probability distribution associated with where an electron may be detected.

For example, hydrogen's lowest-energy orbital is:

spherically symmetric

rather than a thin circular path.

---

From Bohr to Quantum Mechanics

Bohr's model was therefore an important transitional model.

The progression was roughly:

Rutherford nuclear model

↓

Bohr quantized energy levels

↓

wave-particle ideas

↓

modern quantum mechanics

The central idea of quantized atomic energies survived.

The idea of simple fixed circular electron orbits did not.

---

What Bohr Got Right

Several key ideas remain important:

• atomic energies are quantized
• atoms have discrete energy states
• photons are emitted or absorbed during transitions
• photon energy corresponds to an energy difference
• hydrogen's energy levels follow a definite pattern

These ideas became part of modern quantum theory.

---

What Bohr Got Wrong or Oversimplified

The simple Bohr model incorrectly or incompletely treats electrons as:

• particles following fixed circular paths
• having precisely defined classical-style orbits
• behaving independently in multi-electron atoms

Modern quantum mechanics provides a more accurate description.

---

Bohr's Model and Scientific Progress

Bohr's model provides an excellent example of how scientific models develop.

Rutherford's model explained:

nuclear scattering

but could not explain:

atomic stability or line spectra

Bohr added:

quantized electron states

His model successfully explained hydrogen.

Later experiments revealed additional limitations.

Quantum mechanics then provided a more complete theory.

Therefore:

Rutherford → Bohr → quantum mechanics

illustrates how scientific models are refined as new evidence appears.

---

Models Can Be Useful Without Being Perfect

Bohr's model is still widely taught because it provides a useful introduction to:

• energy levels
• electron transitions
• emission
• absorption
• ionization
• atomic spectra

A model does not have to describe every aspect of reality perfectly to be useful.

The important question is:

What can the model explain accurately, and where does it stop working?

---

Worked Example: Absorption or Emission?

An electron changes from:

n = 2 → n = 5

The electron moves to a higher energy state.

Therefore energy must be:

absorbed

Now consider:

n = 5 → n = 2

The electron moves to a lower energy state.

Therefore energy is:

emitted as a photon

A useful rule is:

up → absorb

down → emit

---

Worked Example: Comparing Transitions

Consider two transitions:

A: n = 3 → n = 2

B: n = 4 → n = 2

The n = 4 → n = 2 transition involves a larger energy difference.

Therefore its photon has:

• greater energy
• higher frequency
• shorter wavelength

Remember:

E = hf = hc/λ

---

Worked Example: Energy Level

Find the energy of the n = 3 level in hydrogen.

Use:

Eₙ = −13.6/n² eV

E₃ = −13.6/3²

E₃ = −13.6/9

E₃ ≈ −1.51 eV

---

Worked Example: Transition Energy

Find the photon energy for:

n = 4 → n = 2

First calculate:

E₄ = −13.6/16

E₄ = −0.85 eV

E₂ = −13.6/4

E₂ = −3.40 eV

Photon energy:

E_photon = E₄ − E₂

= −0.85 − (−3.40)

= 2.55 eV

Because the electron moves downward:

a 2.55 eV photon is emitted.

---

Reading an Energy-Level Diagram

When interpreting an atomic energy-level diagram:

Upward arrow

Means:

absorption

Downward arrow

Means:

emission

Longer energy jump

Means:

larger ΔE

and therefore:

higher photon frequency

and:

shorter photon wavelength

This gives the relationship:

larger ΔE → larger E_photon → larger f → smaller λ

---

Common Mistakes

Mistake: Saying Bohr discovered the nucleus.
Rutherford's work established the nuclear model. Bohr introduced quantized atomic energy states.

Mistake: Saying electrons can exist anywhere between Bohr energy levels.
Only certain energies are allowed in the model.

Mistake: Saying electrons continuously lose energy while remaining in an allowed Bohr orbit.
Bohr specifically proposed stable states that do not continuously radiate.

Mistake: Saying an electron emits a photon when moving upward.
An upward transition requires energy absorption.

Mistake: Saying a downward transition absorbs energy.
A downward transition releases energy.

Mistake: Saying every electron transition produces visible light.
Transitions can produce ultraviolet, visible, or infrared radiation.

Mistake: Saying all energy levels are equally spaced.
Hydrogen's levels become closer together as n increases.

Mistake: Treating Bohr's circular electron orbits as the modern atomic model.
Modern quantum mechanics uses orbitals and probability distributions.

Mistake: Saying Bohr's model accurately describes every atom.
It works especially well for hydrogen and other one-electron systems but fails for more complicated atoms.

---

Key Terms

Bohr model — atomic model proposing that electrons occupy specific quantized energy states.

Quantization — restriction of a physical quantity to particular discrete values.

Energy level — allowed energy state of an electron in an atom.

Principal quantum number — integer used to identify Bohr energy levels.

Ground state — lowest possible energy state of an atom.

Excited state — atomic state with energy greater than the ground state.

Electron transition — change of an electron from one allowed energy state to another.

Photon — quantum of electromagnetic radiation.

Emission — release of a photon when an atom moves to a lower energy state.

Absorption — uptake of photon energy that can move an atom to a higher energy state.

Line spectrum — pattern of discrete wavelengths emitted or absorbed by atoms.

Ionization — removal of an electron from an atom.

Ionization energy — minimum energy required to remove an electron from a specified state.

Electron volt (eV) — energy unit commonly used in atomic physics; 1 eV = 1.60 × 10⁻¹⁹ J.

Bohr radius — characteristic radius of hydrogen's ground state in the Bohr model, approximately 5.29 × 10⁻¹¹ m.

Orbit — defined electron path used in the Bohr model.

Orbital — quantum state associated with an electron probability distribution in modern quantum mechanics.

---

Key Takeaways

• Bohr built on Rutherford's nuclear model of the atom.
• Bohr proposed that electrons can occupy only certain allowed energy states.
• These allowed energies are quantized.
• The lowest allowed state is the ground state.
• Higher-energy states are excited states.
• In the original Bohr model, allowed energies were associated with specific electron orbits.
• Electrons can move between allowed energy levels.
• Upward transitions require energy absorption.
• Downward transitions produce photon emission.
• Photon energy equals the difference between atomic energy levels.
• ΔE = hf
• Because c = fλ, larger energy transitions produce shorter-wavelength photons.
• Bohr's model successfully explained hydrogen's line spectrum.
• Hydrogen's energy levels follow Eₙ = −13.6/n² eV.
• Ground-state hydrogen has an energy of −13.6 eV.
• The ground-state ionization energy of hydrogen is 13.6 eV.
• Bohr's model works particularly well for hydrogen and other one-electron systems.
• It does not accurately describe complex multi-electron atoms.
• It cannot fully explain fine spectral structure and several effects caused by external fields.
• Modern quantum mechanics replaces fixed electron orbits with quantum states and orbitals.
• Bohr's model remains valuable because its concept of quantized atomic energies was fundamentally important to the development of quantum physics.

Check Your Understanding

1. What major problem with Rutherford's atomic model did Bohr attempt to solve?
2. What does it mean to say that atomic energy is quantized?
3. What values can the principal quantum number n take?
4. What is the ground state of an atom?
5. What is an excited state?
6. What happens when an electron moves from a lower energy level to a higher one?
7. What happens when an electron moves from a higher energy level to a lower one?
8. State the relationship between photon energy and frequency.
9. Explain why hydrogen produces a line spectrum rather than a continuous spectrum.
10. Calculate the energy of the n = 2 state of hydrogen using Eₙ = −13.6/n² eV.
11. Calculate the energy of the n = 4 state.
12. Calculate the photon energy for a hydrogen transition from n = 4 to n = 2.
13. Which produces a higher-frequency photon: n = 3 → n = 2 or n = 5 → n = 2? Explain.
14. What is the ionization energy of ground-state hydrogen?
15. Explain one major strength of Bohr's atomic model.
16. Why does Bohr's model work particularly well for hydrogen?
17. Give two observations that the simple Bohr model cannot fully explain.
18. Explain the difference between an orbit and an orbital.
19. Why is the Bohr model still useful even though it is not the modern atomic model?
20. Explain how the progression from Rutherford's model to Bohr's model and then to quantum mechanics demonstrates the development of scientific theories.

3. Wave Nature of Matter

Wave Nature of Matter

For centuries, scientists often treated matter and waves as fundamentally different things.

Matter, such as an electron, was pictured as a particle with:

• mass
• momentum
• position
• velocity

Waves were described using properties such as:

• wavelength
• frequency
• diffraction
• interference

The development of quantum physics revealed that this distinction is not always valid.

Particles such as electrons can produce distinctly wave-like behaviour.

This idea is called the:

wave nature of matter

---

From Light to Matter

Scientists first discovered that light could behave in apparently contradictory ways.

Light produces wave phenomena such as:

• interference
• diffraction
• polarization

But experiments such as the photoelectric effect showed that light can also behave as discrete packets called:

photons

A photon has energy:

E = hf

and momentum:

p = h/λ

This led to an important question:

If waves such as light can behave like particles, could particles such as electrons also behave like waves?

---

de Broglie's Hypothesis

In 1924, French physicist Louis de Broglie proposed that particles have an associated wavelength.

He suggested:

λ = h/p

where:

• λ = de Broglie wavelength
• h = Planck's constant
• p = momentum

For a non-relativistic particle:

p = mv

Therefore:

λ = h/mv

This is the:

de Broglie equation

---

Planck's Constant

Planck's constant is:

h = 6.63 × 10⁻³⁴ J·s

It is an extremely small number.

This is one reason matter-wave effects are usually noticeable only for very small particles such as:

• electrons
• neutrons
• atoms
• molecules

For ordinary macroscopic objects, the de Broglie wavelength is usually far too small to observe directly.

---

Momentum and Wavelength

The de Broglie equation is:

λ = h/p

Therefore:

wavelength is inversely proportional to momentum

If momentum increases:

p ↑ → λ ↓

If momentum decreases:

p ↓ → λ ↑

For a particle of constant mass:

higher speed → greater momentum → shorter wavelength

and:

lower speed → smaller momentum → longer wavelength

---

Mass and Wavelength

Using:

λ = h/mv

for particles moving at the same speed:

greater mass → shorter wavelength

Therefore, very massive objects have extraordinarily small matter wavelengths.

This helps explain why quantum wave behaviour is much easier to observe for electrons than for everyday objects.

---

Worked Example: Electron Wavelength

An electron travels at:

2.0 × 10⁶ m/s

Electron mass:

m = 9.11 × 10⁻³¹ kg

Use:

λ = h/mv

First calculate momentum:

p = mv

p = (9.11 × 10⁻³¹)(2.0 × 10⁶)

p = 1.82 × 10⁻²⁴ kg·m/s

Now calculate wavelength:

λ = (6.63 × 10⁻³⁴)/(1.82 × 10⁻²⁴)

λ ≈ 3.64 × 10⁻¹⁰ m

Therefore:

λ ≈ 0.364 nm

This wavelength is comparable with distances between atoms in crystals.

That fact is extremely important for electron diffraction.

---

Worked Example: Increasing Speed

Suppose the same electron doubles its speed.

Originally:

λ = h/mv

If:

v → 2v

then:

λ → λ/2

Therefore, doubling the speed approximately halves the de Broglie wavelength, provided the speed remains low enough that relativistic corrections are unnecessary.

---

Worked Example: Finding Momentum

A particle has a de Broglie wavelength of:

5.0 × 10⁻¹⁰ m

Find its momentum.

Use:

λ = h/p

Rearrange:

p = h/λ

p = (6.63 × 10⁻³⁴)/(5.0 × 10⁻¹⁰)

p ≈ 1.33 × 10⁻²⁴ kg·m/s

---

Matter Waves

The wave associated with a particle is often called a:

matter wave

or:

de Broglie wave

This does not mean an electron is simply a tiny vibrating water wave.

Matter waves are quantum-mechanical.

Their wave behaviour is connected to quantities such as:

• probability
• momentum
• interference
• diffraction

This eventually led to the development of the quantum mechanical:

wavefunction

---

What Would Demonstrate Wave Behaviour?

If electrons behave like waves, they should be capable of producing phenomena normally associated with waves.

Two especially important phenomena are:

diffraction

and:

interference

Therefore, scientists searched for evidence that beams of electrons could produce diffraction patterns.

---

Diffraction

Diffraction is the spreading or bending of waves when they encounter:

• openings
• edges
• regularly spaced structures

Strong diffraction effects occur when the dimensions of the structure are comparable to the wavelength.

For electrons, atomic crystals provide extremely small regularly spaced structures.

Their atomic spacing is often around:

10⁻¹⁰ m

This is comparable to electron de Broglie wavelengths.

---

Crystals as Diffraction Structures

A crystal contains atoms arranged in a regular repeating pattern called a:

crystal lattice

The spacing between atomic planes can be comparable with the wavelength of moving electrons.

Therefore, a crystal can act somewhat like a three-dimensional diffraction structure for electron matter waves.

If electrons were purely classical particles, this wave-like diffraction behaviour would not be expected.

---

Davisson-Germer Experiment

In 1927, Clinton Davisson and Lester Germer performed one of the most famous experiments confirming de Broglie's hypothesis.

They directed a beam of electrons toward a:

nickel crystal

The electrons scattered from the crystal surface.

Instead of producing only the distribution expected for classical particles, the electrons produced strong intensity maxima at particular angles.

---

What Davisson and Germer Observed

The scattered electrons produced a pattern consistent with:

diffraction

Certain angles produced strong electron signals.

Other angles produced weaker signals.

This behaviour was similar to interference and diffraction produced by waves.

The measured wavelength agreed with the wavelength predicted using:

λ = h/p

This provided powerful experimental evidence for:

de Broglie's matter waves

---

G. P. Thomson's Experiment

At around the same time, George Paget Thomson performed another important experiment.

He directed electrons through thin metal foils.

The electrons produced:

diffraction rings

on a detector.

These rings closely resemble diffraction patterns produced by other types of waves.

The results provided further evidence that:

electrons possess wave properties

---

An Interesting Historical Connection

J. J. Thomson had earlier helped establish the electron as a:

particle

His son, G. P. Thomson, later demonstrated the electron's:

wave behaviour

Both descriptions turned out to be important.

This became part of the broader concept of:

wave-particle duality

---

Bragg's Law and Electron Diffraction

Diffraction from crystal planes can be described using:

Bragg's law

nλ = 2d sinθ

where:

• n = diffraction order
• λ = wavelength
• d = spacing between crystal planes
• θ = Bragg angle

Electron diffraction therefore allows the wavelength of electrons to be measured experimentally.

The result can then be compared with:

λ = h/p

Agreement between these two approaches supports de Broglie's hypothesis.

---

Worked Example: Electron Diffraction

Suppose first-order diffraction occurs from crystal planes separated by:

d = 0.20 nm

at an angle:

θ = 30°

For first order:

n = 1

Use:

nλ = 2d sinθ

λ = 2(0.20 nm)sin30°

Since:

sin30° = 0.5

λ = 2(0.20)(0.5)

λ = 0.20 nm

The measured electron wavelength is therefore:

2.0 × 10⁻¹⁰ m

This can be compared with the wavelength predicted from the electron's momentum.

---

Interference

Matter waves can also produce:

interference

When wave amplitudes combine, they can produce:

• constructive interference
• destructive interference

Constructive interference increases detection probability in some locations.

Destructive interference decreases it in others.

This produces characteristic patterns.

---

Electron Double-Slit Experiment

One of the most striking demonstrations of matter waves involves electrons passing through two narrow slits.

If electrons behaved only like classical particles, we might expect two main regions behind the two slits.

Instead, after many electrons are detected, an:

interference pattern

can appear.

This is characteristic of wave behaviour.

---

Individual Electron Detections

An especially important feature of the double-slit experiment is that electrons can be sent through the apparatus:

one at a time

Each electron is detected at a particular location.

This appears particle-like.

However, after many detections, the accumulated pattern reveals:

wave-like interference

This demonstrates that quantum objects do not fit neatly into classical categories.

---

Particle Behaviour

Electrons show particle-like properties because they can:

• carry electric charge
• possess momentum
• transfer energy in localized interactions
• produce individual detector events
• collide with other particles

For example, an electron detector records an electron at a particular location rather than detecting a tiny continuous classical wave spread across the entire screen.

---

Wave Behaviour

Electrons show wave-like properties because they can produce:

• diffraction
• interference
• wavelength-dependent patterns

Their wavelength is related to momentum through:

λ = h/p

Therefore, neither the simple classical particle picture nor the simple classical wave picture is sufficient on its own.

---

Wave-Particle Duality

The idea that quantum objects can display both particle-like and wave-like behaviour is called:

wave-particle duality

Light can display:

wave behaviour + particle behaviour

Matter can also display:

particle behaviour + wave behaviour

This was one of the major conceptual changes introduced by quantum physics.

---

Classical Particle vs Quantum Particle

A classical particle is usually imagined as:

• occupying a definite position
• following a definite trajectory
• possessing definite momentum
• moving along a predictable path

Quantum particles cannot always be described this way.

Instead, quantum mechanics uses a:

wavefunction

to describe the quantum state.

---

The Wavefunction

The wavefunction is usually represented by:

ψ

The wavefunction itself is not directly interpreted as an ordinary physical wave like a water wave.

Instead:

|ψ|²

is related to the probability density for detecting a particle.

This connects wave behaviour with the probability of particle detection.

---

Probability Waves

In quantum mechanics, an electron's wave behaviour helps determine:

where the electron is likely to be detected

For example, an interference experiment produces regions where:

• detection is more likely
• detection is less likely

The individual detection remains localized.

But the distribution of many detections follows a wave-like pattern.

---

Why We Do Not See Matter Waves for Everyday Objects

Every object has a de Broglie wavelength.

This includes:

• electrons
• atoms
• people
• cars
• footballs
• planets

But for macroscopic objects, the wavelength is extraordinarily small.

---

Worked Example: Football

Suppose a football has:

m = 0.50 kg

v = 20 m/s

Momentum:

p = mv

p = (0.50)(20)

p = 10 kg·m/s

Now:

λ = h/p

λ = (6.63 × 10⁻³⁴)/10

λ = 6.63 × 10⁻³⁵ m

This wavelength is unimaginably small.

It is far too small to produce observable diffraction with ordinary structures.

Therefore, a football behaves extremely well as a:

classical particle

---

Electron vs Football

Compare:

Electron

Possible wavelength:

~10⁻¹⁰ m

This is comparable with atomic spacing.

Diffraction can therefore be observed.

Football

Possible wavelength:

~10⁻³⁵ m

No practical slit or crystal structure is remotely small enough to reveal such diffraction.

Therefore:

quantum behaviour is not absent from large objects—it is simply normally unobservable at everyday scales.

---

Accelerating Electrons Through a Voltage

Electrons are often accelerated using an electric potential difference.

If an electron begins approximately from rest and is accelerated through voltage V:

electrical energy gained = kinetic energy

Therefore:

eV = ½mv²

where e is the elementary charge.

Since:

p = mv

we can write:

KE = p²/(2m)

Therefore:

p = √(2meV)

Substitute this into de Broglie's equation:

λ = h/√(2meV)

This is useful for electron diffraction experiments.

---

Worked Example: Electron Accelerated Through a Voltage

Suppose an electron is accelerated through:

150 V

Use:

λ = h/√(2meV)

where:

h = 6.63 × 10⁻³⁴ J·s

m = 9.11 × 10⁻³¹ kg

e = 1.60 × 10⁻¹⁹ C

V = 150 V

This gives approximately:

λ ≈ 1.00 × 10⁻¹⁰ m

or:

λ ≈ 0.100 nm

Again, this is comparable with atomic spacings in crystals.

---

Voltage and Electron Wavelength

From:

λ = h/√(2meV)

we can see:

λ ∝ 1/√V

Therefore:

higher accelerating voltage → greater momentum → shorter wavelength

For example, if the voltage is multiplied by 4:

λ becomes half as large

provided relativistic effects remain negligible.

---

Electron Microscopes

The small de Broglie wavelength of fast electrons has an important practical application:

electron microscopy

Optical microscopes use visible light.

Visible wavelengths are approximately:

400–700 nm

Electrons can have much shorter de Broglie wavelengths.

Shorter wavelengths can potentially reveal much smaller structures.

This allows electron microscopes to achieve extremely high resolution.

---

Why Wavelength Affects Resolution

Diffraction limits the ability of imaging systems to distinguish nearby objects.

In general:

shorter wavelength → potentially better resolution

Electron microscopes therefore use the wave nature of electrons to investigate structures much smaller than those easily resolved using visible-light microscopes.

---

Electron Diffraction in Materials Science

Electron diffraction is also used to investigate:

• crystal structures
• atomic spacing
• material phases
• crystal orientation
• defects

The diffraction pattern contains information about the arrangement of atoms inside the material.

Thus de Broglie's hypothesis became an important practical tool as well as a fundamental quantum idea.

---

Neutron Diffraction

Electrons are not the only particles that show wave behaviour.

Neutrons also have de Broglie wavelengths.

Neutron diffraction can be used to study:

• crystal structures
• magnetic structures
• atomic positions

This confirms that matter waves are not unique to electrons.

---

Atoms and Molecules

Experiments have demonstrated interference and diffraction using increasingly large particles, including:

• atoms
• molecules

This reinforces the idea that the de Broglie relationship applies generally to matter.

The transition between obvious quantum behaviour and everyday classical behaviour is therefore not simply:

small things quantum, large things classical

The situation also depends on interactions with the environment and the ability to maintain quantum coherence.

---

de Broglie and the Bohr Model

de Broglie's idea also helped explain why Bohr's allowed electron states might occur.

Imagine an electron's matter wave fitting around an orbit.

For a stable standing wave:

circumference = whole number of wavelengths

Therefore:

2πr = nλ

where:

n = 1, 2, 3, ...

Only certain wavelengths fit exactly.

This provides a wave-based way of understanding why only certain states are allowed.

---

Standing Waves and Quantization

Consider a wave around a circular path.

If the circumference contains:

1 wavelength

or:

2 wavelengths

or:

3 wavelengths

the wave can join smoothly with itself.

But a non-integer number of wavelengths would not form the same stable standing-wave pattern.

This connects:

wave behaviour → allowed states → quantization

The modern quantum model is more sophisticated than this simple picture, but the standing-wave analogy is useful.

---

From de Broglie to Quantum Mechanics

de Broglie's hypothesis had a major influence on the development of quantum theory.

If particles have wave properties, scientists needed an equation describing those waves.

This helped inspire Erwin Schrödinger to develop:

wave mechanics

The resulting Schrödinger equation became one of the foundations of quantum mechanics.

---

Particle and Wave Descriptions

Quantum mechanics incorporates aspects of both descriptions.

---

Neither Classical Picture Is Complete

It is tempting to ask:

"Is an electron really a particle or really a wave?"

Quantum physics shows that neither classical picture completely describes an electron.

An electron is a:

quantum object

Depending on the experiment, its behaviour may reveal characteristics that resemble classical:

• particles
• waves

The experimental arrangement determines which properties can be observed.

---

Matter Waves Are Not Ordinary Waves

Matter waves should not be confused with:

• sound waves
• water waves
• waves on strings

These classical waves involve physical disturbances in a medium or field.

Quantum matter waves describe the state and probability behaviour of particles.

This is why the modern interpretation is more subtle than saying:

"an electron is literally a tiny wave."

---

Evidence for de Broglie's Hypothesis

Strong evidence includes:

Electron diffraction

Electrons produce diffraction patterns when scattered from crystals.

Electron interference

Electrons can form interference patterns.

Agreement with λ = h/p

Experimentally measured electron wavelengths agree with de Broglie's prediction.

Neutron and atom diffraction

Other particles also show wave-like behaviour.

Electron microscopy

The wave properties of electrons are routinely exploited in technology.

---

A Powerful Scientific Prediction

de Broglie's hypothesis is an important example of scientific reasoning.

The logic was approximately:

light behaves as waves

but:

light also behaves as particles

therefore perhaps:

matter behaves as particles

but could also:

behave as waves

This was a bold theoretical prediction.

Experiments later confirmed it.

That sequence:

hypothesis → prediction → experiment → evidence

is a powerful example of how physics develops.

---

Worked Example: Comparing Two Electrons

Electron A has momentum:

p

Electron B has momentum:

2p

For electron A:

λ_A = h/p

For electron B:

λ_B = h/(2p)

Therefore:

λ_B = λ_A/2

Electron B has twice the momentum and half the wavelength.

---

Worked Example: Comparing Two Particles

Particle A:

m = m

v = v

Particle B:

m = 2m

v = v

Since:

λ = h/mv

Particle B has twice the momentum.

Therefore:

λ_B = λ_A/2

The more massive particle has the shorter de Broglie wavelength when both move at the same speed.

---

Worked Example: Finding Speed

An electron has wavelength:

4.0 × 10⁻¹⁰ m

Find its speed.

Use:

λ = h/mv

Rearrange:

v = h/(mλ)

Substitute:

v = (6.63 × 10⁻³⁴) / [(9.11 × 10⁻³¹)(4.0 × 10⁻¹⁰)]

v ≈ 1.82 × 10⁶ m/s

---

Problem-Solving Strategy

For de Broglie calculations:

Identify what is known

Look for:

• mass
• velocity
• momentum
• wavelength
• accelerating voltage

Calculate momentum if necessary

For non-relativistic motion:

p = mv

Use de Broglie's equation

λ = h/p

or:

λ = h/mv

Check units

Momentum:

kg·m/s

Wavelength:

m

Check reasonableness

Small particles often have measurable wavelengths.

Large everyday objects should produce extremely tiny wavelengths.

---

Common Mistakes

Mistake: Saying only light has wave-particle duality.
Matter particles such as electrons also exhibit wave behaviour.

Mistake: Saying de Broglie's equation is λ = hp.
The correct relationship is λ = h/p.

Mistake: Saying greater momentum gives a greater wavelength.
Wavelength is inversely proportional to momentum.

Mistake: Forgetting to calculate momentum first.
For non-relativistic motion, use p = mv.

Mistake: Thinking electron diffraction proves electrons are ordinary classical waves.
Electrons are quantum objects exhibiting wave-like diffraction while still producing localized detections.

Mistake: Saying electrons disappear into waves.
Individual electrons are still detected as localized events.

Mistake: Saying matter waves are only theoretical.
Electron, neutron, atom, and molecular diffraction provide experimental evidence.

Mistake: Saying everyday objects have no de Broglie wavelength.
They do, but the wavelength is extraordinarily small.

Mistake: Assuming λ = h/mv is valid at all speeds.
At speeds approaching the speed of light, relativistic momentum must be used.

---

Key Terms

de Broglie hypothesis — proposal that matter has an associated wavelength determined by its momentum.

de Broglie wavelength — wavelength associated with a particle, given by λ = h/p.

Matter wave — quantum wave associated with a material particle.

Momentum — quantity of motion; for non-relativistic particles, p = mv.

Planck's constant — fundamental constant h = 6.63 × 10⁻³⁴ J·s.

Wave-particle duality — quantum principle that matter and radiation can display both particle-like and wave-like behaviour.

Diffraction — spreading or interference of waves caused by interaction with structures comparable in size to their wavelength.

Interference — combination of waves producing regions of reinforcement and cancellation.

Crystal lattice — regular repeating arrangement of atoms in a crystal.

Davisson-Germer experiment — experiment demonstrating electron diffraction from a nickel crystal.

Bragg's law — relationship nλ = 2d sinθ used to describe diffraction from crystal planes.

Wavefunction — mathematical description of a quantum state.

Probability density — quantity related to the probability of detecting a quantum particle in a particular region.

Electron diffraction — wave-like scattering of electrons that produces diffraction patterns.

Standing wave — wave pattern produced when waves combine so that stable nodes and antinodes form.

Quantum object — object described by quantum mechanics rather than completely by classical particle or wave models.

---

Key Takeaways

• de Broglie proposed that matter has wave properties.
• The de Broglie wavelength is λ = h/p.
• For non-relativistic particles, λ = h/mv.
• Greater momentum produces a shorter de Broglie wavelength.
• Greater speed produces a shorter wavelength for a particle of fixed mass.
• Greater mass produces a shorter wavelength for particles travelling at the same speed.
• Electrons can have wavelengths comparable with atomic spacings in crystals.
• Crystal lattices can therefore produce electron diffraction.
• The Davisson-Germer experiment provided strong evidence for electron matter waves.
• Electron diffraction measurements agree with de Broglie's predicted wavelength.
• Electrons can also produce interference patterns.
• Individual electrons are detected at localized positions even though many detections form wave-like patterns.
• This behaviour is part of wave-particle duality.
• Matter waves are not ordinary mechanical waves.
• Modern quantum mechanics connects the wavefunction with probabilities of particle detection.
• All matter has a de Broglie wavelength.
• Macroscopic objects have wavelengths so tiny that their wave behaviour is normally impossible to observe.
• Electron wavelengths can be much shorter than visible-light wavelengths.
• Electron microscopy uses this property to achieve very high resolution.
• de Broglie's hypothesis helped lead to the development of modern quantum mechanics.

Check Your Understanding

1. State de Broglie's hypothesis.
2. Write the equation for de Broglie wavelength.
3. What does p represent in λ = h/p?
4. How does wavelength change when particle momentum increases?
5. Calculate the momentum of an electron moving at 3.0 × 10⁶ m/s.
6. Use your answer to Question 5 to calculate its de Broglie wavelength.
7. What happens to an electron's wavelength if its momentum doubles?
8. What happens to its wavelength if its momentum is halved?
9. Why are matter-wave effects easier to observe for electrons than for footballs?
10. What is diffraction?
11. Why are crystals useful for demonstrating electron diffraction?
12. Describe the Davisson-Germer experiment.
13. What observation from the experiment provided evidence for electron wave behaviour?
14. How did the measured electron wavelength support de Broglie's hypothesis?
15. Explain how the electron double-slit experiment demonstrates both particle-like and wave-like behaviour.
16. What is meant by wave-particle duality?
17. Calculate the de Broglie wavelength of a 0.20 kg ball travelling at 5.0 m/s.
18. Explain why the wave nature of the ball in Question 17 would not normally be observable.
19. Why can electron microscopes achieve much higher resolution than ordinary optical microscopes?
20. Compare the classical particle description and wave description of an electron, and explain why modern quantum mechanics requires a description that goes beyond either classical picture alone.

4. Quantum Mechanical Model

Quantum Mechanical Model

The quantum mechanical model of the atom is the modern scientific model used to describe electrons in atoms.

Unlike the Bohr model, it does not picture electrons as tiny particles travelling around the nucleus in fixed circular paths.

Instead, quantum mechanics describes electrons using:

• wavefunctions
• quantized energy states
• atomic orbitals
• probability distributions

The central idea is:

We generally describe where an electron is likely to be detected rather than assigning it a fixed classical orbit around the nucleus.

---

From Bohr to Quantum Mechanics

Bohr's model was a major improvement over Rutherford's model.

It introduced:

quantized energy levels

and successfully explained much of the hydrogen spectrum.

However, the Bohr model pictured electrons travelling along specific:

orbits

This idea eventually proved inadequate.

Experiments involving:

• electron diffraction
• atomic spectra
• multi-electron atoms
• magnetic fields
• electron wave behaviour

required a more sophisticated theory.

This became:

quantum mechanics

---

Wave Nature of Electrons

Louis de Broglie proposed that particles such as electrons have an associated wavelength:

λ = h/p

where:

• λ = de Broglie wavelength
• h = Planck's constant
• p = momentum

Experiments confirmed that electrons can produce:

• diffraction
• interference

Therefore, electrons cannot be described completely as simple classical particles.

---

Schrödinger's Contribution

In 1926, Austrian physicist Erwin Schrödinger developed an equation that describes the wave-like behaviour of quantum systems.

It became known as the:

Schrödinger equation

Its solutions provide information about the allowed quantum states of electrons.

For atoms, these solutions lead to the concept of:

atomic orbitals

---

The Wavefunction

A quantum state can be represented mathematically by a:

wavefunction

usually written:

ψ

The wavefunction contains information about the quantum state.

However, ψ itself is not simply interpreted as the physical position of the electron.

Instead:

|ψ|²

is related to the:

probability density

for detecting the electron.

---

Probability Density

Probability density tells us how likely an electron is to be detected in a particular region of space.

A region with:

large |ψ|²

has a greater probability density.

A region with:

small |ψ|²

has a lower probability density.

This does not mean the electron is physically smeared through the atom like a cloud of material.

The "cloud" represents:

probability

---

Electron Probability Clouds

Electron probability distributions are often represented visually using:

electron clouds

In such a diagram:

• denser regions represent greater probability
• less dense regions represent lower probability
• the nucleus is located near the center

The cloud does not represent a fuzzy physical substance.

It represents where repeated measurements are more or less likely to detect the electron.

---

Probability and Measurement

Suppose scientists prepared many identical hydrogen atoms in the same quantum state.

If they measured the electron's position repeatedly, they would not necessarily obtain exactly the same position every time.

Instead, the results would form a distribution.

After many measurements:

measurement locations → probability distribution

The electron cloud represents this distribution.

---

Atomic Orbitals

An atomic orbital is a quantum state associated with a particular electron probability distribution around the nucleus.

Orbitals have characteristic:

• energies
• shapes
• orientations

Common orbital types include:

• s orbitals
• p orbitals
• d orbitals
• f orbitals

---

Orbit vs Orbital

These two words sound similar but have very different meanings.

Orbit

Used in the simplified Bohr model.

An orbit is pictured as:

a definite path followed by an electron

Orbital

Used in quantum mechanics.

An orbital describes:

a quantum state with a characteristic probability distribution

Therefore:

orbit ≠ orbital

---

The 1s Orbital

The lowest-energy state of hydrogen is the:

1s orbital

It has a:

spherical probability distribution

around the nucleus.

The probability density is greatest near the nucleus and generally decreases with increasing distance.

There is no thin circular track showing where the electron travels.

Instead, there is a three-dimensional probability distribution.

---

s Orbitals

s orbitals are spherically symmetric.

Examples include:

• 1s
• 2s
• 3s

As the principal quantum number increases, the orbital generally becomes larger and more complex.

---

p Orbitals

p orbitals have a different shape.

They are commonly represented as two-lobed structures.

For a p subshell there are three orientations:

• pₓ
• pᵧ
• p_z

These point along different spatial axes.

The probability distribution therefore has both:

shape and orientation

---

d and f Orbitals

Higher-energy atoms also involve:

d orbitals

and:

f orbitals

These have more complicated probability distributions.

The increasing complexity of these orbitals helps quantum mechanics explain:

• electron configurations
• periodic table structure
• chemical bonding
• magnetic behaviour
• atomic spectra

---

Orbitals Are Three-Dimensional

Atomic orbitals are not flat diagrams.

They describe probability distributions in:

three-dimensional space

A textbook drawing is only a visual representation.

For example, the two lobes of a p orbital represent regions associated with the electron's quantum state, not two physical containers holding pieces of an electron.

---

Quantized Energy Still Exists

Quantum mechanics did not discard Bohr's idea that atomic energies are quantized.

Electrons still occupy:

discrete allowed energy states

However, modern quantum mechanics provides a much more complete description of those states.

Bohr:

energy levels + fixed circular orbits

Quantum mechanics:

energy levels + wavefunctions + orbitals + probabilities

---

Principal Energy Levels

Electron states can be organized partly using the:

principal quantum number, n

Possible values are:

n = 1, 2, 3, 4, ...

Larger values of n generally correspond to:

• higher energy
• larger characteristic distance from the nucleus

But unlike the Bohr model, each principal level can contain different:

sublevels and orbitals

---

Sublevels

Principal energy levels contain sublevels commonly labelled:

s, p, d, f

For example:

n = 1 contains:

1s

n = 2 contains:

2s and 2p

n = 3 contains:

3s, 3p and 3d

n = 4 contains:

4s, 4p, 4d and 4f

This structure is much richer than Bohr's simple circular shells.

---

Number of Orbitals

Each type of sublevel contains a particular number of orbitals.

Each individual orbital can contain a maximum of:

2 electrons

with opposite spin states.

---

Quantum Numbers

Electron states can be described using a set of:

quantum numbers

These provide information about an electron's quantum state.

The four commonly introduced quantum numbers are:

• principal quantum number, n
• orbital angular momentum quantum number, l
• magnetic quantum number, mₗ
• spin quantum number, mₛ

Together, they provide a much more detailed description than simply stating an electron's Bohr orbit.

---

Principal Quantum Number

The principal quantum number:

n

is associated with the main energy level.

Values are:

1, 2, 3, 4, ...

It is related to the characteristic size and energy of the orbital.

---

Orbital Angular Momentum Quantum Number

The quantum number:

l

is associated with orbital type.

Common correspondence:

l = 0 → s

l = 1 → p

l = 2 → d

l = 3 → f

This helps determine the shape of the orbital.

---

Magnetic Quantum Number

The magnetic quantum number:

mₗ

describes the possible orientations associated with an orbital.

For example, the p sublevel has three orbitals.

These correspond to different spatial orientations commonly represented as:

• pₓ
• pᵧ
• p_z

---

Electron Spin

Electrons also possess an intrinsic quantum property called:

spin

The spin quantum number can take values:

mₛ = +½

or:

mₛ = −½

These are often described informally as:

spin up

and:

spin down

Electron spin is a quantum property and should not be imagined literally as a tiny ball physically rotating around its axis.

---

Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons in the same atom can have exactly the same set of four quantum numbers.

One important consequence is:

each orbital can contain a maximum of two electrons

and those electrons have opposite spin states.

This principle is essential for understanding electron configurations and the periodic table.

---

Nodes

Some orbitals contain regions where the probability density is:

zero

These regions are called:

nodes

An electron in that quantum state will not be detected at a node.

Nodes are a natural consequence of the wave nature of electrons.

They are similar in concept to nodes in standing waves.

---

Standing Waves

The wave nature of electrons provides a useful analogy.

A vibrating string can form only certain stable standing-wave patterns.

Similarly, only certain electron wavefunctions satisfy the conditions required for stable atomic states.

This helps explain why atomic energies are:

quantized

---

Quantization from Wave Behaviour

In the Bohr model, quantized orbits were largely introduced as a rule.

Quantum mechanics provides a deeper explanation.

Only certain wavefunctions satisfy the physical conditions around the nucleus.

Therefore:

only certain quantum states are allowed

and those states have particular:

energies

This makes quantization a natural result of the mathematical theory rather than simply an added assumption.

---

The Uncertainty Principle

Quantum mechanics also introduces the:

Heisenberg uncertainty principle

It states that position and momentum cannot both be specified with unlimited precision.

A simplified form is:

ΔxΔp ≥ ħ/2

where:

• Δx = uncertainty in position
• Δp = uncertainty in momentum
• ħ = h/(2π)

This is a fundamental quantum relationship, not simply a limitation of poor measuring instruments.

---

Why Fixed Electron Orbits Are a Problem

A classical orbit assumes that an electron has a precisely defined:

• position
• momentum
• trajectory

at every moment.

Quantum mechanics does not generally allow that classical description.

Instead, electron states are represented using:

wavefunctions and probabilities

This is one reason the fixed-orbit picture is replaced by orbitals.

---

Electron Cloud vs Fixed Orbit

---

Interpreting an Electron Cloud

Suppose an electron cloud diagram contains many dots.

A common interpretation is:

more dots in a region → greater probability of detecting the electron there

It does not mean:

the electron spends exactly that percentage of time as a classical particle moving through that region

The quantum interpretation concerns the probabilities of measurement outcomes.

---

Probability Does Not Mean "We Just Don't Know"

In classical physics, probability can sometimes reflect missing information.

For example, if we do not know exactly how a coin was thrown, we might describe its outcome probabilistically.

Quantum probability is more fundamental.

The quantum state itself determines probabilities for possible measurement outcomes.

Quantum mechanics does not simply say:

"the electron has a definite classical orbit but we don't know which one."

That hidden classical-orbit picture does not reproduce quantum behaviour.

---

Electron Transitions

Electrons can still transition between quantum energy states.

If an atom changes from a higher energy state to a lower one:

photon emitted

with:

E_photon = ΔE

and:

E = hf

Therefore:

ΔE = hf

This explains discrete atomic emission spectra.

---

Absorption

An atom can absorb a photon if its energy matches an allowed energy difference.

Therefore:

photon absorbed → atom moves to higher-energy state

Because only certain transitions are possible, atoms absorb specific wavelengths.

This produces:

absorption spectra

---

Atomic Spectra

Quantum mechanics explains why each element has a characteristic spectrum.

Different atoms have different:

• nuclear charges
• numbers of electrons
• electron-electron interactions
• allowed quantum states

Therefore, they have different energy differences.

Different energy differences produce different:

photon wavelengths

This gives each element its characteristic spectral pattern.

---

Why Quantum Mechanics Improves on Bohr

Bohr's model works very well for hydrogen.

However, atoms such as:

• helium
• carbon
• oxygen
• sodium
• iron

contain several electrons.

These electrons interact with:

• the nucleus
• one another

Quantum mechanics can incorporate these interactions far more successfully than the simple Bohr model.

---

Multi-Electron Atoms

In a multi-electron atom, electron-electron repulsion changes the energies of electron states.

This produces:

• sublevels
• shielding effects
• complex spectra
• electron configurations

The simple formula:

Eₙ = −13.6/n² eV

works for hydrogen but cannot describe ordinary multi-electron atoms in that simple form.

---

Explaining the Periodic Table

The quantum mechanical model helps explain the structure of the:

periodic table

Electron configurations determine many chemical properties.

For example:

• elements in the same group often have similar outer-electron configurations
• orbital filling helps explain periods
• d orbitals help explain transition metals
• f orbitals help explain lanthanides and actinides

The quantum model therefore connects:

physics → atomic structure → chemistry

---

Explaining Chemical Bonds

Chemical bonding depends strongly on electron wavefunctions and orbitals.

Quantum mechanics helps explain:

• covalent bonds
• molecular orbitals
• bond energies
• molecular shapes
• electron sharing

The Bohr model cannot provide a complete explanation of these phenomena.

---

Explaining Spectral Details

The quantum mechanical model can explain features that the original Bohr model could not adequately describe, including:

• multi-electron spectra
• fine structure
• effects of magnetic fields
• effects of electric fields
• electron spin

These provide important evidence supporting the modern model.

---

Magnetic Effects

When atoms are placed in magnetic fields, spectral lines can split.

This is associated with the:

Zeeman effect

Quantum mechanics explains this using properties such as:

• angular momentum
• magnetic quantum numbers
• electron spin

The simple Bohr model cannot fully account for these observations.

---

The Shape of Orbitals

Different orbitals arise because different mathematical solutions of the Schrödinger equation have different spatial forms.

For example:

s → spherical

p → two-lobed

d → more complex

The shapes are therefore not arbitrary artistic diagrams.

They represent mathematical properties of quantum states.

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Probability Surfaces

Orbital diagrams often draw a boundary around a region containing a high percentage of the probability of finding the electron.

This boundary is sometimes called a:

probability surface

The electron is not absolutely confined inside that surface.

There can still be some probability of detecting it outside.

This is important when interpreting orbital diagrams.

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Most Probable Does Not Mean Certain

Suppose an orbital diagram shows a region where the electron is highly likely to be detected.

That does not mean:

the electron must always be inside that region

Quantum mechanics predicts probabilities.

A high-probability region is:

likely

not:

guaranteed

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Radial Probability

For spherical states such as hydrogen's 1s orbital, scientists can calculate the probability of finding the electron at different distances from the nucleus.

Interestingly, the point of greatest probability density and the most probable radial distance are not exactly the same concept.

This illustrates why quantum probability distributions are more sophisticated than simply drawing a fuzzy version of a Bohr orbit.

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Quantum Mechanics and Chemistry

The quantum model explains why electrons fill atoms in structured ways.

For example:

Hydrogen:

1s¹

Helium:

1s²

Lithium:

1s² 2s¹

Carbon:

1s² 2s² 2p²

Oxygen:

1s² 2s² 2p⁴

These electron configurations help explain the chemical behaviour of the elements.

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Worked Example: Identifying an Orbital

An electron is described as being in a:

2p orbital

What can we infer?

The number:

2

indicates the principal energy level:

n = 2

The letter:

p

indicates the orbital type.

A p orbital has a characteristic two-lobed probability distribution.

Therefore, 2p describes a:

quantum state

not a circular path at a fixed distance from the nucleus.

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Worked Example: Probability

Suppose two regions of an electron cloud are compared.

Region A has a high probability density.

Region B has a low probability density.

This means:

the electron is more likely to be detected in Region A than Region B

It does not mean that part of the electron is permanently stored in each region.

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Worked Example: Electron Transition

An electron changes between two atomic states with an energy difference:

ΔE = 4.0 × 10⁻¹⁹ J

Find the frequency of the photon.

Use:

ΔE = hf

Rearrange:

f = ΔE/h

f = (4.0 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 6.03 × 10¹⁴ Hz

Therefore, the photon frequency is approximately:

6.0 × 10¹⁴ Hz

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Worked Example: Finding Wavelength

Using:

c = fλ

Rearrange:

λ = c/f

λ = (3.00 × 10⁸)/(6.03 × 10¹⁴)

λ ≈ 4.98 × 10⁻⁷ m

or:

498 nm

This photon lies in the visible region of the electromagnetic spectrum.

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Comparing Rutherford, Bohr, and Quantum Models

Each model represented an important stage in the development of atomic theory.

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Why Earlier Models Were Still Important

Rutherford's model established the existence of the:

nucleus

Bohr introduced:

quantized atomic energy

de Broglie introduced:

matter waves

Schrödinger developed:

wave mechanics

Heisenberg helped establish:

uncertainty and quantum mechanics

Modern atomic theory developed from all of these contributions.

Scientific models evolve as new experimental evidence becomes available.

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Why the Quantum Model Is More Accurate

The quantum mechanical model successfully explains a much wider range of observations.

These include:

• atomic stability
• quantized energy levels
• hydrogen spectra
• multi-electron spectra
• orbital shapes
• electron configurations
• chemical bonding
• periodic trends
• magnetic properties
• electron diffraction
• fine spectral structure

It also makes extremely precise quantitative predictions that agree with experiments.

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Accuracy Does Not Mean Easy to Visualize

The Bohr model is easy to picture:

tiny electrons orbiting a nucleus

The quantum model is harder to visualize because nature at atomic scales does not behave like familiar everyday objects.

A model being easy to visualize does not necessarily make it more accurate.

Quantum mechanics relies heavily on:

mathematics + probability + experimental evidence

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Classical Thinking vs Quantum Thinking

Classically, we might ask:

"Where exactly is the electron, and what exact path is it following?"

Quantum mechanically, a more appropriate question is often:

"What is the probability of detecting the electron in a particular region?"

This represents one of the biggest conceptual changes from classical to quantum physics.

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Common Mistakes

Mistake: Saying an orbital is the same as an orbit.
An orbit is a fixed path in the Bohr model. An orbital is a quantum state associated with a probability distribution.

Mistake: Saying an electron cloud is a cloud of electron material.
It represents probability density.

Mistake: Saying electrons move around inside orbitals like planets inside containers.
Orbitals describe quantum states, not physical containers.

Mistake: Saying quantum mechanics abandoned quantized energy.
Quantization remains fundamental and is explained more deeply by quantum mechanics.

Mistake: Saying the electron is definitely located where the cloud is darkest.
A darker or denser region represents greater probability, not certainty.

Mistake: Saying an electron has no position.
An electron can be detected at a position. Quantum mechanics predicts probabilities for possible measurement outcomes.

Mistake: Saying uncertainty results only from poor equipment.
The uncertainty principle is a fundamental feature of quantum mechanics.

Mistake: Drawing p orbitals as two electrons.
The two lobes are parts of one orbital's probability distribution.

Mistake: Saying Bohr's model is completely useless.
It remains a useful simplified model for introducing quantized energies and hydrogen spectra.

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Key Terms

Quantum mechanical model — modern model describing atomic electrons using quantum states, wavefunctions, orbitals, and probabilities.

Wavefunction (ψ) — mathematical function describing a quantum state.

Probability density — quantity related to the likelihood of detecting a particle in a particular region.

Atomic orbital — quantum state associated with a characteristic electron probability distribution.

Electron cloud — visual representation of electron probability density.

Orbit — fixed electron path used in the Bohr model.

Ground state — lowest-energy quantum state of an atom.

Excited state — atomic state with energy above the ground state.

Principal quantum number — quantum number associated mainly with principal energy level and orbital size.

Sublevel — subdivision of a principal energy level, labelled s, p, d, or f.

Quantum numbers — values used to describe quantum states of electrons.

Spin — intrinsic quantum property of particles such as electrons.

Pauli exclusion principle — principle stating that no two electrons in an atom can have identical sets of four quantum numbers.

Node — region where a particular wavefunction has zero probability density.

Uncertainty principle — fundamental relationship limiting simultaneous precision in quantities such as position and momentum.

Electron configuration — arrangement of electrons among atomic orbitals.

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Key Takeaways

• The quantum mechanical model is the modern model of atomic structure.
• It replaces Bohr's fixed electron paths with quantum states and atomic orbitals.
• Electrons have wave-like as well as particle-like properties.
• A quantum state is represented mathematically by a wavefunction, ψ.
• |ψ|² is related to electron probability density.
• Electron clouds represent probability distributions, not physical clouds of electron material.
• An orbital is not the same as an orbit.
• Orbitals have characteristic energies, shapes, and orientations.
• Common orbital types are s, p, d, and f.
• s orbitals are spherically symmetric.
• p orbitals have characteristic two-lobed probability distributions.
• Orbitals can contain regions of zero probability called nodes.
• Atomic energies remain quantized in the quantum model.
• Quantization arises naturally from the allowed solutions of quantum mechanics.
• Electron states can be described using quantum numbers.
• Each orbital can contain a maximum of two electrons with opposite spin states.
• The uncertainty principle prevents a fully classical description involving precise electron trajectories.
• Quantum mechanics explains multi-electron atoms much more successfully than the Bohr model.
• It helps explain atomic spectra, electron configurations, periodic trends, chemical bonding, and magnetic behaviour.
• The Bohr model remains useful as a simplified model, but the quantum mechanical model agrees with a much wider range of experimental evidence.
• The progression from Rutherford to Bohr to quantum mechanics demonstrates how scientific models become more sophisticated as new evidence is discovered.

5. Uncertainty and Probability