7.1 Quantum Theory of Light
In classical physics, light was regarded strictly as a continuous electromagnetic wave produced by accelerating electric charges. However, classical theories failed to explain phenomena such as black body radiation curves and the line spectra emitted by atoms. To resolve these inconsistencies, quantum physics emerged, proposing that energy is emitted and absorbed in discrete packets called quanta or photons. Furthermore, the concept of wave-particle duality demonstrated that light possesses particle properties, while microscopic particles such as electrons exhibit wave characteristics.
Learning Objectives
- Explain the concept of black bodies and black body radiation.
- Describe the historical development from classical theory to quantum theory through the contributions of key physicists.
- Compare continuous spectrum and line spectrum.
- Explain the concept of quantum of energy and calculate photon energy using \(E = hf\) and \(E = \frac{hc}{\lambda}\).
- Determine the power of a light source and the number of photons emitted per second using \(P = nhf = \frac{nhc}{\lambda}\).
- Explain wave-particle duality and calculate the de Broglie wavelength using \(\lambda = \frac{h}{p} = \frac{h}{mv}\).
Graph of Black Body Radiation Intensity vs Wavelength & Wave-Particle Duality Summary
Illustrate how black body emission curves shift to shorter wavelengths at higher temperatures and summarize the key dual properties of waves and particles.
Black Body Radiation Curves
As temperature rises, the peak radiation intensity shifts to shorter wavelengths. The graph is narrower on the left side, where wavelengths are short and frequencies are high.
Wave-Particle Duality Summary
Light and matter both show wave-like and particle-like behaviour. The de Broglie wavelength links momentum to wave behaviour.
de Broglie wavelength
\(\lambda = \dfrac{h}{p} = \dfrac{h}{mv}\)where \(p\) is momentum, \(m\) is mass and \(v\) is velocity.
Wave properties
Diffraction and interference are observed for light and electrons.
Particle properties
Momentum, collisions and discrete photon energy are observed.
Short Explanation
What is a Black Body?
A black body is an idealised body that is able to absorb all electromagnetic radiation that falls on it. It emits thermal radiation depending on its temperature.
Black Body Radiation
The radiation emitted forms a continuous spectrum and is unaffected by the nature of the black body surface. An object emitting electromagnetic radiation determined by its temperature is called a black body radiator.
Quantum of Energy
Energy is emitted and absorbed in discrete packets called quanta. The energy of each quantum is directly proportional to wave frequency: \(E \propto f\).
Photon Energy
A photon is a discrete packet of light energy. Its energy is \(E = hf = \frac{hc}{\lambda}\).
Photon energy and power
\(E = hf = \dfrac{hc}{\lambda}\)\(P = nhf = \dfrac{nhc}{\lambda}\)
where \(h = 6.63 \times 10^{-34}\ \mathrm{J\,s}\), \(c = 3.00 \times 10^8\ \mathrm{m\,s^{-1}}\), \(f\) is frequency, \(\lambda\) is wavelength and \(n\) is the number of photons emitted per second.
de Broglie wavelength
\(\lambda = \dfrac{h}{p} = \dfrac{h}{mv}\)where \(p = mv\) is momentum, \(m\) is mass and \(v\) is velocity.
Historical Development of Quantum Theory
Max Planck
Introduced the concept of quantum, stating that electromagnetic waves emitted by a black body are discrete. Energy in each quantum is directly proportional to wave frequency.
Albert Einstein
Introduced the photon concept. Photon energy is directly proportional to light wave frequency. Successfully explained the photoelectric effect.
Niels Bohr
Explained the hydrogen line spectrum. Electrons orbit in specific shells; electron transitions from higher to lower energy shells emit photons.
Louis de Broglie
Introduced the hypothesis of the wave nature of particles. Postulated wave-particle duality for light and subatomic particles.
Continuous Spectrum vs Line Spectrum
| Feature | Continuous Spectrum | Line Spectrum |
|---|---|---|
| Production | Dispersion of white light through a prism | Excited atoms |
| Appearance | Unbroken sequence of colours with no gaps | Series of discrete coloured lines with unique wavelengths and frequencies |
| Visible range | 400 nm to 750 nm | Specific wavelengths depending on element |
| Use | Shows all visible wavelengths | Can identify the presence of elements |
Mercury Vapour Lamp Line Spectrum
| Colour | Frequency, \(f\) / Hz | Photon energy, \(E\) / J |
|---|---|---|
| Violet | \(7.41 \times 10^{14}\) | \(4.91 \times 10^{-19}\) |
| Blue | \(6.88 \times 10^{14}\) | \(4.56 \times 10^{-19}\) |
| Green | \(5.49 \times 10^{14}\) | \(3.65 \times 10^{-19}\) |
| Yellow-orange | \(5.19 \times 10^{14}\) | \(3.44 \times 10^{-19}\) |
Try to Answer First
Answer in your mind, then press “Check Answer”.
What is the energy of a light photon with a wavelength of \(400\ \mathrm{nm}\)? Given \(c = 3.00 \times 10^8\ \mathrm{m\,s^{-1}}\), \(h = 6.63 \times 10^{-34}\ \mathrm{J\,s}\).
Check Answer
State the mathematical relationship between the momentum of a particle and its de Broglie wavelength.
Check Answer
Why can we not observe the wave properties of a moving baseball?
Check Answer
Common Mistakes
- Forgetting to convert nanometres to metres using \(1\ \mathrm{nm} = 10^{-9}\ \mathrm{m}\), leading to an error by a factor of \(10^9\).
- Confusing photon energy \(E = hf\) with light beam power \(P = nhf\), omitting the number of photons per second \(n\).
- Assuming that higher temperature black bodies peak at longer wavelengths, whereas higher temperatures shift the peak to shorter wavelengths.
- Forgetting that de Broglie wavelength applies to moving particles \(p = mv\), not stationary particles.
Concept Misunderstandings
A black body must physically look black in colour under room light.
A black body is an idealised physical body that absorbs all incident radiation regardless of frequency. When heated, it emits thermal radiation across all wavelengths and can glow red, white, or blue.
Light is purely a wave and electrons are purely particles.
Both light and matter have wave-particle duality. Light shows particle properties such as discrete photon packets and momentum, while electrons show wave properties such as diffraction and interference.
The energy of a photon increases when the intensity or brightness of the light source increases.
Photon energy depends solely on its frequency \(E = hf\). Increasing light intensity increases the number of photons emitted per second, not the energy of individual photons.
Summary
- A black body absorbs all incident electromagnetic radiation and emits continuous thermal radiation depending solely on its temperature.
- Classical physics predicted that radiation intensity continuously increases with frequency, failing to match experimental curves at short wavelengths.
- Max Planck resolved this by introducing discrete quanta of energy, \(E = hf\).
- A photon is a discrete packet of light energy whose energy is \(E = \frac{hc}{\lambda}\).
- Total power emitted is \(P = nhf = \frac{nhc}{\lambda}\), where \(n\) is the number of photons emitted per second.
- Louis de Broglie established wave-particle duality, where a moving particle exhibits a wavelength \(\lambda = \frac{h}{mv}\). Electron diffraction confirmed this wave nature.
Short Activity
Objective: Test your understanding of quantum theory, photon energy and wave-particle duality.
A. Objective Quiz
1 According to Planck's quantum theory, how does the energy of a photon change when the frequency of the light wave increases?
2 Which experimental observation confirmed the wave properties of electrons in 1927?
B. Fill in the Blanks
3 An idealised body that absorbs all electromagnetic radiation falling on it is called a .
4 The formula for the de Broglie wavelength of a particle with mass \(m\) and velocity \(v\) is \(\lambda =\) .
C. Matching / Drag and Drop
Match each physicist to their correct contribution to quantum physics. If using a phone, tap the answer first, then tap the matching box.
