Quantized Energy

Quantized Energy Recall how we described light as a wave and used a wave model to explain many phenomenon including diffraction and refraction. The wave model works for these situations but it doesn’t always work …

Incandescent Emission Spectrum

This graph describes how light is emitted from an incandescent body (something that emits light when heated). It cannot be described with the wave model of light but can be described when restricting the energy of the vibrating atom to integer multiples of a fixed amount. 𝐸=𝑛ℎ𝑓 Where 𝑛 is an integer number, 𝑓 is the frequency, and ℎ is Planck’s constant: 6.626× 10 −34 Js

The Photoelectric Effect Another case where the wave model fails. When radiation (i.e. light) falls on an object, electrons are emitted/ejected only once a certain threshold energy is reached by the radiation. The energy of the radiation is directly proportional to its frequency, therefore there is a threshold frequency for this to occur. The threshold energy is known as the work function.

Einstein showed, in 1905, that to explain the photoelectric effect radiation must consist of discrete quantized bundles of energy we now call photons. The energy of a photon can be found from: Where 1 eV is the energy needed to accelerate an electron across a 1 V potential difference. 1 eV=1.60× 10 −19 J 𝐸=ℎ𝑓= ℎ𝑐 𝜆 = 1240 eV∙nm 𝜆

Kinetic Energy of an Emitted Electron Once the threshold energy is reached, all extra energy goes towards the kinetic energy of the emitted electron. Where f0 is the threshold frequency. 𝐾𝐸=ℎ𝑓−ℎ 𝑓 0

Example 1 A sodium cathode with a threshold wavelength of 536 nm is hit by 348 nm light. What is the velocity of the ejected electrons? ( 𝑚 𝑒 =9.11× 10 −31 kg)

The Compton Effect 𝑝= ℎ𝑓 𝑐 = ℎ 𝜆 Where 𝑝 is the momentum of a photon. Recall that 𝐸 𝑡 =𝑭∙𝒗= Δ𝒑 𝑡 ∙𝒗 so it is quite natural to assume that if light has energy, it must have momentum as well. Einstein, in 1916, showed that the momentum of a photon should be equal to 𝐸 𝑐 and so 𝑝= ℎ𝑓 𝑐 = ℎ 𝜆 Where 𝑝 is the momentum of a photon.

As such, one can expect that light behaves in collisions very much like billiard balls. In 1922, American physicist Arthur Holly Compton observed that X rays scattered by a graphite target increased in wavelength and thus lost both energy and momentum. This is known as the Compton effect. Compton also later observed the conservation of momentum in collisions between photons and electrons.