7.1 - Quantum Theory of Light

Black Bodies and Thermal Radiation

The electromagnetic spectrum includes gamma rays, X-rays, ultraviolet radiation, visible light, infrared radiation, microwaves and radio waves. Wavelength increases and frequency decreases in this order.
All objects emit electromagnetic radiation. The radiation emitted depends on the object’s temperature; hotter objects can emit higher-frequency radiation such as visible light and ultraviolet radiation.
Thermal radiation includes visible and invisible electromagnetic radiation, such as infrared radiation.
A black body is an ideal body that absorbs all electromagnetic radiation incident on it. Its emitted radiation forms a continuous spectrum, depends on temperature and is independent of the nature of its surface.
A black body radiator emits electromagnetic radiation according to its temperature.
Light entering the ear cavity undergoes repeated reflections and absorption by the inner walls; the cavity acts like a black body.
A hotter black body emits thermal radiation at all wavelengths. Its spectrum curve is narrower on the short-wavelength, high-frequency side.
On a graph of radiation intensity against wavelength, increasing temperature shifts the maximum intensity towards a shorter wavelength. Higher light intensity means more light energy is emitted.
The illustrated spectra compare 4 000 K red stars, 5 270 K white stars such as the Sun, and 7 000 K blue stars. Everyday illustrations include a thermographic car image, an oven heating element and light bulbs at different temperatures.

Origin and Development of Quantum Theory

Light is produced by vibrating electric charges. Electrons in a hot object vibrate rapidly and randomly; higher temperature makes their vibrations more energetic and increases the emitted light.
Classical theory assumes that electrons vibrating at the same frequency have the same energy content. It treats the frequency range as continuous and unrestricted, predicting that the light energy could increase without limit at high frequencies.
Black body experiments show that intensity falls beyond the peak towards shorter wavelengths, instead of continuing to increase with frequency. This disagreement prompted quantum theory.

Physicist	Contribution and limitation
Isaac Newton (1704)	Described light as a stream of particles or corpuscles; his comparison of light speeds in glass and air failed to explain refraction correctly.
Thomas Young (1801)	The double-slit experiment demonstrated light’s wave properties, but classical wave theory could not explain the black body radiation spectrum.
John Dalton	Modelled matter as indivisible atoms, with the same type of atoms in the same element; could not explain atomic light spectra.
J. J. Thomson (1897)	Discovered negatively charged electrons and studied their behaviour; could not explain atomic line spectra.
Max Planck (1900)	Proposed discrete energy quanta for black body radiation, with energy proportional to frequency; radiation intensity is low at high frequencies.
Albert Einstein (1905)	Proposed photons with energy proportional to frequency, explaining photoelectric observations that classical theory could not explain.
Niels Bohr	Explained hydrogen’s line spectrum using specific electron shells; an electron moving from a higher to a lower energy shell emits a photon.
Louis de Broglie (1924)	Proposed that particles have wave properties. His ideas and Einstein’s led to wave-particle duality for light and subatomic particles.

Continuous Spectrum, Line Spectrum and Energy Quanta

A prism disperses white light into a continuous visible spectrum, approximately 400–750 nm, with no separating gaps between colours.
An excited atom produces a line spectrum containing separate coloured lines at distinctive wavelengths and frequencies. Each element has its own pattern, allowing its presence to be identified.
The hot mercury vapour lamp gives these values:

Colour	Frequency / $1 0^{14} Hz$	Energy / $1 0^{- 19} J$
Violet	7.41	4.91
Blue	6.88	4.56
Green	5.49	3.65
Yellow-orange	5.19	3.44

Quantum of energy: a discrete energy packet.
Photon: a transferable quantum of light energy. Higher light frequency gives greater photon energy.

c = λ f, E = h f = \frac{h c}{λ}

$E$ : photon energy in J; $f$ : frequency in Hz; $λ$ : wavelength in m.
Planck’s constant, $h = 6.63 \times 1 0^{- 34} J s$ ; speed of light in vacuum, $c = 3.00 \times 1 0^{8} m s^{- 1}$ .
Shorter wavelength gives higher photon energy. For the two illustrated photons:

E_{400 nm} E_{750 nm} = \frac{( 6.63 \times 1 0 ^{- 34} ) ( 3.00 \times 1 0 ^{8} )}{400 \times 1 0 ^{- 9}} = 4.97 \times 1 0^{- 19} J = \frac{( 6.63 \times 1 0 ^{- 34} ) ( 3.00 \times 1 0 ^{8} )}{750 \times 1 0 ^{- 9}} = 2.65 \times 1 0^{- 19} J

Wave-Particle Duality and de Broglie Wavelength

Wave properties include diffraction and interference. Particle properties include momentum, kinetic energy and collisions.
Light and electrons exhibit both types of properties: wave-particle duality. Other electromagnetic radiation and subatomic particles such as protons and neutrons also have this duality.
de Broglie’s hypothesis states that all particles can exhibit wave properties:

λ = \frac{h}{p}, p = m v, λ = \frac{h}{m v}

$p$ : particle momentum; $m$ : particle mass; $v$ : particle velocity.
Greater momentum gives a shorter wavelength. Since $h$ is extremely small, massive particles have wavelengths too short for their wave properties to be observed readily.
Electron diffraction experiments in 1927 confirmed de Broglie’s hypothesis. Electrons passing through thin graphite form a diffraction pattern resembling red laser light passing through a circular aperture.
An electron beam’s wavelength is approximately 1 000–10 000 times shorter than light’s wavelength, enabling higher magnification in an electron microscope. Red blood cell images illustrate the difference from an optical microscope.

Photon Energy and Power Calculations

For $n$ photons emitted per second, the light power is:

P = nh f = \frac{nh c}{λ}, n = \frac{P λ}{h c}

$P$ is in W and $n$ is in $s^{- 1}$ .
50 W red lamp, $λ = 7.0 \times 1 0^{- 7} m$ :

n = \frac{50 ( 7.0 \times 1 0 ^{- 7} )}{( 6.63 \times 1 0 ^{- 34} ) ( 3.00 \times 1 0 ^{8} )} = 1.76 \times 1 0^{20} s^{- 1}

670 nm red laser pen, $n = 3.37 \times 1 0^{18} s^{- 1}$ :

P = \frac{( 3.37 \times 1 0 ^{18} ) ( 6.63 \times 1 0 ^{- 34} ) ( 3.00 \times 1 0 ^{8} )}{6.7 \times 1 0 ^{- 7}} = 1.00 W

100 W bulb emitting $2.92 \times 1 0^{19}$ photons per second with 10% of its output power: use $P = 10 W$ .

λ = \frac{nh c}{P} = \frac{( 2.92 \times 1 0 ^{19} ) ( 6.63 \times 1 0 ^{- 34} ) ( 3.00 \times 1 0 ^{8} )}{10} = 5.81 \times 1 0^{- 7} m = 581 nm