Similarities Between Light and Matter Revision Notes for VCE SSCE Physics

Similarities Between Light and Matter

Introduction to quantum physics

Quantum physics is the physics of the very small - the physics of atoms, photons, electrons, and other fundamental particles. Before the early 20th century, humanity had no understanding of quantum physics, but today around 35% of the economy in modern countries relies on quantum technology.

Everyday applications of quantum physics include:

MRI scans in medical diagnosis
GPS navigation systems
Digital photography
Mobile phones and internet
Laser scanners at checkouts

At the heart of quantum physics is the concept of quantisation - the idea that physical quantities such as charge, mass, energy, and momentum can only have certain discrete values. These quantities cannot take values between the allowed values. This is fundamentally different from classical physics, where quantities can vary continuously.

infoNote

Quantisation means that physical quantities come in discrete "packets" rather than varying smoothly. Think of it like climbing stairs versus walking up a ramp - you can only stand on specific steps, not between them.

The momentum of photons

Photons carry momentum

Although photons have no mass, they do carry momentum. This seems impossible according to classical physics, where momentum is a property of moving masses. However, Maxwell predicted that electromagnetic waves carried momentum, and Einstein showed that Planck's law of blackbody radiation required light quanta (photons) to carry momentum.

The momentum of a photon is given by:

$p = \frac{h}{\lambda}$

where:

$p$ = momentum (kg $\cdot$ m $\cdot$ s $^{-1}$ )
$h$ = Planck's constant ( $6.63 \times 10^{-34}$ J·s)
$\lambda$ = wavelength (m)

chatImportant

Despite having zero mass, photons carry momentum. This counterintuitive fact is a fundamental prediction of quantum physics and has been confirmed by countless experiments.

Real-world examples of photon momentum

Comet tails:

Comets have two tails as they orbit the Sun:

The ion tail is caused by the stream of charged particles from the Sun transferring momentum to the comet's gases
The dust tail is caused by photons from the Sun transferring their momentum to the dust and gas released from the comet

Both tails point away from the Sun due to momentum transfer.

Space sails:

Some spacecraft use 'space sails' for propulsion. The mechanism works as follows:

Photons from the Sun hit the sail and reflect
Before reflection, a photon has momentum $\frac{h}{\lambda}$ (to the right)
After reflection, it has momentum $\frac{h}{\lambda}$ (to the left)
Each reflecting photon transfers momentum $\Delta p = \frac{2h}{\lambda}$ to the sail
If $n$ photons reflect per second, the force on the sail is $F = \frac{2nh}{\lambda}$

infoNote

The factor of 2 in the momentum transfer arises because the photon's momentum changes from $+\frac{h}{\lambda}$ to $-\frac{h}{\lambda}$ , giving a total change of $\frac{2h}{\lambda}$ . This is similar to a ball bouncing off a wall - the wall experiences twice the momentum of the ball.

Conservation of momentum

Momentum is conserved in all interactions between photons and other particles, including electrons. This principle is fundamental to understanding photon behaviour.

lightbulbExample

Worked Example: Calculating thrust on a space sail

A large space sail reflects $10^{21}$ photons per second, each with wavelength 550 nm. The thrust can be calculated as:

$F = \frac{2nh}{\lambda \Delta t}$

$F = \frac{2(10^{21})(6.63 \times 10^{-34})}{(550 \times 10^{-9})(1)}$

$F = 2.3 \times 10^{-6} \text{ N}$

Although this force is extremely small, in the vacuum of space with no friction, even tiny forces can accelerate spacecraft over long periods.

Compton scattering

In 1923, Compton investigated collisions between X-ray photons and electrons in graphite. He found that:

The scattered photons had longer wavelengths (less energy)
The collisions were inelastic
Conservation of energy and momentum could only be satisfied if photons had momentum given by $p = \frac{h}{\lambda}$

This provided strong experimental evidence that photons carry momentum.

chatImportant

Compton scattering was a landmark experiment that conclusively demonstrated the particle nature of light. It showed that photons collide with electrons like billiard balls, exchanging both energy and momentum according to the laws of conservation.

lightbulbExample

Worked Example: Photon-electron collision

An X-ray photon with momentum $6.6 \times 10^{-23}$ kg $\cdot$ m $\cdot$ s $^{-1}$ collides head-on with a stationary electron. The electron gains momentum of $1.1 \times 10^{-22}$ kg $\cdot$ m $\cdot$ s $^{-1}$ in the same direction. Using conservation of momentum:

$P_{\text{before}} = P_{\text{after}}$

$(6.6 \times 10^{-23}) = (1.1 \times 10^{-22}) + P_{\text{photon after}}$

$P_{\text{photon after}} = -4.4 \times 10^{-23} \text{ kg} \cdot \text{m} \cdot \text{s}^{-1}$

The negative sign indicates the photon bounces back in the opposite direction.

Emission spectra

Continuous vs. line spectra

All objects above absolute zero (0 K) emit electromagnetic radiation. The type of spectrum emitted depends on the nature of the source.

Continuous spectrum:

A tungsten filament incandescent light globe produces a continuous spectrum:

All wavelengths are present with no gaps
The spectrum is close to the blackbody spectrum of a perfect emitter
Only wavelengths from approximately 400 nm to 700 nm are visible to the human eye
Infrared wavelengths (> 700 nm) can be felt but not seen
Ultraviolet wavelengths (< 400 nm) are also invisible

The continuous spectrum can be observed using a spectroscope, which separates wavelengths using a prism or diffraction grating.

Emission line spectrum:

When an electric current passes through a gas discharge tube containing hydrogen, a completely different spectrum is observed:

Only specific wavelengths are present
The spectrum consists of discrete lines, not a continuous range
The visible part of the hydrogen spectrum (called the Balmer series) shows lines at:
- 656.2 nm (red)
- 486.1 nm (blue-green)
- 434.0 nm (blue-violet)
- 410.1 nm (violet)

The energies of these photons are related to their wavelengths by:

$E = hf = \frac{hc}{\lambda}$

where:

$E$ = photon energy (J or eV)
$h$ = Planck's constant
$f$ = frequency (Hz)
$c$ = speed of light ( $3.00 \times 10^8$ m $\cdot$ s $^{-1}$ )
$\lambda$ = wavelength (m)

infoNote

Unlike continuous spectra, the energies emitted in line spectra do not depend on the temperature of the gas. This is because line spectra arise from specific energy transitions within atoms, not from thermal vibrations.

Unique spectra for each element

Different elements produce different emission line spectra. For example, helium produces a spectrum with major peaks at wavelengths including 389 nm, 502 nm, 588 nm, 668 nm, and 706 nm.

Each element has its own unique 'fingerprint' spectrum. This allows us to:

Identify elements present in a sample
Detect elements in stars and comets
Analyse the atmospheres of exoplanets

infoNote

In fact, helium was first discovered in the Sun's atmosphere (during a solar eclipse) before it was found on Earth. The element is named after "Helios," the Greek word for Sun!

Metal vapour lamps

Metals are not gases at room temperature, so they must be vaporised to observe their spectra. Metal vapour lamps (such as mercury and sodium lamps) work by:

Creating a hot electric arc
Vaporising the metal
Exciting atoms into higher energy states
Producing photons as atoms return to lower energy states

Polar auroras

Polar auroras (aurora borealis in the northern hemisphere and aurora australis in the southern hemisphere) are natural examples of emission spectra. The process involves:

High-speed charged particles from the solar wind are channelled by Earth's magnetic fields near the poles
These particles strike atoms in the upper atmosphere
The atoms are excited into high energy states
As they decay, they emit photons
Different colours correspond to different atoms - mainly oxygen and nitrogen

infoNote

The spectacular colours of auroras are emission line spectra in action. Green and red auroras come from oxygen atoms, while blue and purple come from nitrogen. The specific colours depend on which energy transitions occur in these atoms.

Energy states of atoms

Quantisation of energy states

In 1913, Niels Bohr proposed that electrons in atoms can only occupy certain fixed energy states. Light is emitted when an electron moves from one state to a lower state. This is another example of quantisation - a phenomenon that cannot be explained by classical physics.

The energy difference between states is given by:

$\Delta E = hf = \frac{hc}{\lambda}$

where $\Delta E$ is the energy difference between the two states.

Energy state diagrams

Energy states of atoms are represented using energy level diagrams. For hydrogen:

The vertical axis shows energy in electron volts (eV)
Horizontal lines represent the allowed energy states
Each state is labelled with a quantum number $n$
The ground state (lowest energy) has quantum number $n = 1$
Higher energy states have $n = 2, 3, 4$ , etc.
The ionisation energy (13.6 eV for hydrogen) represents the energy needed to completely remove an electron

When an atom receives enough energy, an electron can transition to a higher energy state. If the atom receives 13.6 eV or more (for hydrogen), it loses an electron and becomes an ion (a charged atom).

chatImportant

The ground state is the most stable configuration of an atom. All higher energy states are unstable and will eventually decay back to the ground state, emitting photons in the process.

Transitions and photon emission

States above the ground state are unstable. Electrons in these excited states decay to lower energy states until reaching the ground state. Each transition involves:

A loss of energy equal to the energy difference between states
Emission of a photon with energy $E = \Delta E$

The diagram shows an example where a hydrogen atom in the $n = 4$ state:

First decays to $n = 2$ , emitting a 2.6 eV photon
Then decays to $n = 1$ (ground state), emitting a 10.2 eV photon

Multiple decay paths

An atom in an excited state can decay through different paths. For example, an atom in the $n = 5$ state could:

Decay directly to $n = 1$ (emitting one high-energy photon)
Decay via $n = 4$ , then $n = 3$ , then $n = 2$ , then $n = 1$ (emitting four lower-energy photons)
Follow any other combination of paths

All possible transitions produce the complete line spectrum we observe from a gas discharge tube. The wavelengths present in the spectrum correspond to all possible energy differences between states.

infoNote

This explains why emission spectra contain multiple lines. Each line corresponds to a different possible transition between energy states. The more complex the atom, the more possible transitions, and the more lines in its spectrum.

Calculating photon energies

lightbulbExample

Worked Example: Finding photon energy from wavelength

Using the helium emission spectrum, what is the energy of the 389 nm photon?

The greatest energy corresponds to the smallest wavelength. Using:

$\Delta E = \frac{hc}{\lambda}$

$\Delta E = \frac{(6.63 \times 10^{-34})(3.00 \times 10^8)}{389 \times 10^{-9}}$

$\Delta E = 5.11 \times 10^{-19} \text{ J}$

$\Delta E = 3.19 \text{ eV}$

(since 1 J = $6.25 \times 10^{18}$ eV)

Using energy state diagrams

Energy state diagrams can be used to:

Identify which transitions produce specific photon energies
Calculate the highest or lowest energy photons that can be emitted
Determine whether absorption of photons at specific wavelengths is possible

lightbulbExample

Worked Example: Sodium energy states

For the sodium energy level diagram shown:

a) Finding the transition for a 1.51 eV photon:

A 1.51 eV photon corresponds to the energy gap between $n = 4$ (3.61 eV) and $n = 2$ (2.10 eV), since $3.61 - 2.10 = 1.51$ eV.

b) Calculating the highest frequency photon:

The highest frequency photon from $n = 4$ to ground state has energy 3.61 eV:

$f = \frac{E}{h} = \frac{3.61}{4.14 \times 10^{-15}} = 8.72 \times 10^{14} \text{ Hz}$

c) Determining if a 591 nm photon can be absorbed:

$E = \frac{hc}{\lambda} = 4.14 \times 10^{-15} \times \frac{3 \times 10^8}{591 \times 10^{-9}} = 2.1 \text{ eV}$

This matches the energy gap from ground state to $n = 2$ , so absorption is possible.

Absorption spectra

What are absorption spectra?

Absorption line spectra occur when photons with specific energies are absorbed from an otherwise continuous spectrum. They appear as dark lines against a coloured background.

A well-known example is the Sun's spectrum, which contains dark lines called Fraunhofer lines. These occur when:

The Sun emits a continuous spectrum of light
The light passes through the cooler gases in the Sun's atmosphere and Earth's atmosphere
Atoms in these atmospheres absorb photons at specific wavelengths
These absorbed photons match energy gaps between atomic energy states
Dark lines appear in the spectrum at these wavelengths

How absorption creates dark lines

When a photon is absorbed:

Its energy must exactly match an energy gap between states
The electron transitions to a higher energy state
The photon is absorbed and then re-emitted in a random direction
Since almost all random directions are not in the original direction, the photon appears to be 'missing' from that wavelength
This creates a dark line in the spectrum

chatImportant

For absorption to occur, the photon energy must exactly match an energy gap between states. If the energy is even slightly different, the photon will not be absorbed. This is why absorption spectra have the same pattern as emission spectra for a given element.

Example: The 656 nm absorption line

This dark line in the solar spectrum is caused by hydrogen atoms in the $n = 2$ state:

They absorb photons of energy 1.9 eV (wavelength 656 nm)
Electrons transition from $n = 2$ to $n = 3$
This matches the energy gap between these states

Alternative energy state representation

Energy state diagrams can also be drawn with the ground state as a negative value:

Ground state ( $n = 1$ ): -13.6 eV
First excited state ( $n = 2$ ): -3.40 eV
Second excited state ( $n = 3$ ): -1.50 eV
Third excited state ( $n = 4$ ): -0.85 eV

The negative values indicate that electrons are bound to the atom. To ionise the atom (remove an electron completely), enough energy must be added to reach 0 eV:

From ground state: needs 13.6 eV
From $n = 2$ state: needs 3.4 eV
From $n = 3$ state: needs 1.5 eV

infoNote

The spacing between states is identical to the other representation - only the reference point has changed. Both representations are valid and useful in different contexts. The negative value representation makes it clear that energy must be added to remove an electron.

Why are atomic energy states stable?

Problems with the classical model

The classical 'planetary' model pictures electrons orbiting the nucleus like planets orbiting the Sun. In this model, the electrostatic force between the positive proton and negative electron keeps the electron in orbit.

However, this classical model has two major problems:

1. Continuous vs. quantised orbits: Classical physics predicts electron orbits could have any energy, like satellites orbiting Earth at different altitudes. But experimental evidence shows electron energies are quantised - only specific values are allowed.

2. Orbital collapse: According to classical electromagnetic theory, accelerating charged particles (like electrons in orbit) should continuously emit radiation and lose energy. This would cause orbits to collapse within a fraction of a second. But atoms are stable - they don't collapse.

chatImportant

The classical model cannot explain why atoms are stable or why energy states are quantised. These phenomena require quantum mechanics to understand properly. The stability of atoms is one of the most fundamental predictions of quantum physics.

The quantum explanation

To explain stable atomic energy states, we must use the wave properties of electrons:

Electrons behave as matter waves, not just as particles
Only certain 'standing wave' patterns fit around the nucleus
These allowed patterns correspond to the quantised energy states
The wave nature prevents orbital collapse

This explanation requires quantum mechanics and cannot be understood using classical physics alone.

Wave-particle duality

Building images photon by photon

When we take photographs, we don't normally think about individual photons. However, images can be built up photon by photon. The figure shows how an image develops from just a few photons per pixel to many thousands. This demonstrates that:

Light is composed of individual photons (particle-like)
Each photon can be detected individually
Large numbers of photons produce the continuous images we're familiar with

Double slit experiments with photons

The double slit experiment has been crucial in understanding the nature of light. When performed at very low light intensities:

With many photons:

A clear interference pattern forms
This is explained by wave superposition
Waves from one slit interfere with waves from the other

With single photons (one at a time):

Individual photons appear to hit random locations
As more photons accumulate, the interference pattern emerges
Each photon seems to 'interfere with itself'
The photon cannot be split - it's detected at one location only

chatImportant

The key observation: Even when only one photon passes through the slits at a time, the interference pattern still appears. This suggests each photon somehow 'knows' about both slits. This is one of the most mysterious and counterintuitive aspects of quantum physics.

When detectors are placed at the slits:

The photon is detected passing through one slit or the other (never both)
The interference pattern disappears
This is particle-like behaviour

Without detection at the slits:

The photon maintains wave-like behaviour
It travels to the screen avoiding dark regions
The interference pattern appears

This demonstrates the dual nature of light - it exhibits both wave-like and particle-like behaviour depending on how it's observed.

infoNote

The act of measurement fundamentally changes the behaviour of photons. This is not because our measuring devices are crude or disturb the photons - it's a fundamental feature of quantum mechanics. The universe itself seems to behave differently depending on whether or not we observe it.

Double slit experiments with electrons

In 1961, Claus Jönsson performed double slit experiments with electrons. In 2013, Roger Bach and his team reduced the electron beam to one electron at a time. The results were identical to single-photon experiments:

Individual electrons created spots on the detector
As more electrons accumulated, an interference pattern emerged
The pattern only formed when electrons were not detected at the slits
When detected at the slits, electrons behaved like particles and no interference pattern formed

This proves electrons also have a dual nature:

Particle-like behaviour when detected/measured
Wave-like behaviour when not detected, showing interference

chatImportant

The double slit experiment with electrons proves that wave-particle duality is not just a property of light - it's a fundamental feature of all matter. Electrons, atoms, and even molecules have been shown to exhibit this dual nature.

Quantisation and light

Einstein's explanation of the photoelectric effect challenged the prevailing wave theory of light and provided an early example of quantisation. He proposed that light consists of discrete packets of energy (quanta, now called photons).

However, this doesn't mean we can ignore wave properties. Single photon diffraction and interference experiments show that until a photon is detected, it retains wave properties. We must describe its position using probabilities, not definite locations.

infoNote

Key point: Light exhibits both particle properties (when detected) and wave properties (when propagating). This is the principle of wave-particle duality.

Quantisation and the nature of atoms

What quantities are quantised?

Several quantities in atoms can only have discrete values:

Electric charge on the nucleus (always a whole number of proton charges)
Number of nucleons (protons and neutrons) in the nucleus
Number of electrons in the atom
Charge on electrons ( $-1.602 \times 10^{-19}$ C) and protons ( $+1.602 \times 10^{-19}$ C)
Energy states of electrons in atoms

Why quantisation matters

The quantisation of energy states is particularly important because:

It arises from the wave properties of electrons
It explains the stable structure of atoms
It accounts for emission and absorption line spectra
It shows that each element has a unique spectral fingerprint
It enables us to identify elements in distant stars and exoplanets

infoNote

The Schrödinger equation of quantum mechanics provides the most accurate explanation of quantised energy states. While simplified models (like electrons as standing waves around nuclei) can help us visualise the concept, they're not completely accurate representations of the complex reality described by quantum mechanics.

Key formulas for photons and matter

It's essential to use the correct formulas when solving problems:

Property	Matter	Photons
Wavelength	$\lambda = \frac{h}{p}$	$\lambda = \frac{h}{p}$
Momentum	$p = mv$	$p = \frac{E}{c}$
Energy/Momentum relation	$p = \sqrt{2mE_k}$	$c = f\lambda$
Kinetic Energy	$E_k = \frac{1}{2}mv^2$	$E = hf = \frac{hc}{\lambda}$

chatImportant

Important notes on Planck's constant:

For calculations in Joules: $h = 6.63 \times 10^{-34}$ J·s
For calculations in electron volts: $h = 4.14 \times 10^{-15}$ eV·s
Use the eV·s value only for $E = hf = \frac{hc}{\lambda}$ when you want energy in eV

Remember!

bookmarkSummary

Key Points to Remember:

Photons carry momentum given by $p = \frac{h}{\lambda}$ , even though they have no mass. Momentum is conserved in all photon interactions.
Emission line spectra consist of discrete wavelengths emitted by excited atoms. Each element has a unique spectrum determined by its energy states. Continuous spectra (like from incandescent bulbs) contain all wavelengths.
Energy states in atoms are quantised - electrons can only exist in specific energy levels. Transitions between states involve emission or absorption of photons with energy $\Delta E = hf = \frac{hc}{\lambda}$ .
Absorption spectra show dark lines where photons with specific energies have been absorbed by atoms, causing upward transitions. The absorbed photons match energy gaps between states.
Wave-particle duality is fundamental to quantum physics. Both light and matter exhibit wave-like behaviour (interference) and particle-like behaviour (discrete detection), depending on how they're measured. Single photon and single electron double slit experiments provide strong evidence for this dual nature.

Similarities Between Light and Matter (VCE SSCE Physics): Revision Notes