DP IB Physics: HL

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IBPhysics: HLDPRevision Notes12. Quantum & Nuclear Physics (HL only)12.1 The Interaction of Matter with Radiation12.1.1 Photons

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First teaching 2014

Last exams 2024

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12.1.1 Photons

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Photons

  • In classical wave theory, electromagnetic (EM) radiation is assumed to behave as a wave
    • This is demonstrated by the fact EM radiation exhibits phenomena such as diffraction and interference

  • However, experiments from the last century, such as discovering the photoelectric effect and atomic line spectra, can only be explained if the EM radiation is thought of as behaving as particles

  • These experiments have formed the basis of quantum theory, which will be explored in detail in this section

Photons

  • Photons are fundamental particles which make up all forms of electromagnetic radiation
  • A photon is defined as

A massless ''packet'' or a ''quantum'' of electromagnetic energy

12-1-1-photons-ib-hl

A wave packet (photon)

  • It is also defined as:

A massless particle that possesses an energy equal to E = hf

  • This means is that the energy of a photon is not transferred continuously, but as discrete packets of energy
  • In other words, each photon carries a specific amount of energy, and transfers this energy all in one go, rather than supplying a consistent amount of energy

  • The energy of a photon can be calculated using the formula:

E italic equals h f

  • Substituting in from the wave equation, energy can also be equal to:

begin mathsize 16px style E italic equals fraction numerator h c over denominator lambda end fraction end style

  • Where:
    • E = energy of the photon (J)
    • h = Planck's constant (J s)
    • c = the speed of light (m s-1)
    • f = frequency in Hertz (Hz)
    • λ = wavelength (m)

  • This equation tells us:
    • The higher the frequency of EM radiation, the higher the energy of the photon
    • The energy of a photon is inversely proportional to the wavelength
    • A long-wavelength photon of light has a lower energy than a shorter-wavelength photon

 

Worked example

Light of wavelength 490 nm is incident normally on a surface, as shown in the diagram.
The power of the light is 3.6 mW. The light is completely absorbed by the surface.

Calculate the number of photons incident on the surface in 2.0 s.

Step 1: Write down the known quantities

    • Wavelength, λ = 490 nm = 490 × 10−9 m
    • Power, P = 3.6 mW = 3.6 × 10−3 W
    • Time, t = 2.0 s

Step 2: Write the equations for wave speed and photon energy

begin mathsize 16px style w a v e space s p e e d colon space space space space space c space equals space f lambda space rightwards arrow space f space equals space c over lambda end style

    size 16px p size 16px h size 16px o size 16px t size 16px o size 16px n size 16px space size 16px e size 16px n size 16px e size 16px r size 16px g size 16px y size 16px colon size 16px space size 16px space size 16px space size 16px space size 16px space size 16px E size 16px space size 16px equals size 16px space size 16px h size 16px f size 16px space size 16px rightwards arrow size 16px space size 16px E size 16px space size 16px equals size 16px space fraction numerator size 16px h size 16px c over denominator size 16px lambda end fraction

Step 3: Calculate the energy of one photon

Step 4: Calculate the number of photons hitting the surface every second

fraction numerator size 16px p size 16px o size 16px w size 16px e size 16px r size 16px space size 16px o size 16px f size 16px space size 16px l size 16px i size 16px g size 16px h size 16px t size 16px space size 16px s size 16px o size 16px u size 16px r size 16px c size 16px e over denominator size 16px e size 16px n size 16px e size 16px r size 16px g size 16px y size 16px space size 16px o size 16px f size 16px space size 16px o size 16px n size 16px e size 16px space size 16px p size 16px h size 16px o size 16px t size 16px o size 16px n end fraction size 16px space size 16px equals fraction numerator size 16px 3 size 16px. size 16px 6 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px minus size 16px 3 end exponent over denominator size 16px 4 size 16px. size 16px 06 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px minus size 16px 19 end exponent end fraction size 16px space size 16px space size 16px equals size 16px space size 16px 8 size 16px. size 16px 9 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px 15 size 16px space size 16px s to the power of size 16px minus size 16px 1 end exponent

Step 5: Calculate the number of photons that hit the surface in 2 s

size 16px left parenthesis size 16px 8 size 16px. size 16px 9 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px 15 size 16px space size 16px cross times size 16px space size 16px 2 size 16px right parenthesis size 16px space size 16px equals size 16px space size 16px 1 size 16px. size 16px 8 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px 16

Photon Momentum

  • Einstein showed that a photon travelling in a vacuum has momentum, despite it having no mass
  • The momentum (p) of a photon is related to its energy (E) by the equation:

  • Where
    • c = the speed of light
    • p = the momentum of the photon (kg m s-1)
    • E = the energy of the photon (J)

Worked example

A 5.0 mW laser beam is incident normally on a fixed metal plate. The cross-sectional area of the beam is 8.0 × 10-6 m2. The light from the laser has frequency 5.6 × 1014 Hz.

Assuming that all the photons are absorbed by the plate, calculate the momentum of the photon, and the pressure exerted by the laser beam on the metal plate.

Step 1: Write down the known quantities

    • Power, P = 5.0 mW = 5.0 × 10−3 W
    • Frequency, f = 5.6 × 1014 Hz
    • Cross-sectional area, A = 8.0 × 10−6 m2

Step 2: Write the equations for photon energy and momentum

begin mathsize 16px style p h o t o n space e n e r g y colon space E space equals space h f p h o t o n space m o m e n t u m colon space p space equals space E over c space equals space fraction numerator h f over denominator c end fraction space end style

Step 3: Calculate the photon momentum

Step 4: Calculate the number of photons incident on the plate every second

      begin mathsize 16px style fraction numerator power space of space light space source over denominator energy space of space one space photon end fraction space equals space fraction numerator 5.0 space cross times space 10 to the power of negative 3 end exponent over denominator h f end fraction end style

begin mathsize 14px style space equals space space fraction numerator 5.0 space cross times space 10 to the power of negative 3 end exponent over denominator left parenthesis 6.63 space cross times space 10 to the power of negative 34 end exponent right parenthesis space cross times space left parenthesis 5.6 space cross times space 10 to the power of 14 right parenthesis end fraction space equals space 1.35 space cross times space 10 to the power of 16 space s to the power of negative 1 end exponent end style

Step 5: Calculate the force exerted on the plate in a 1.0 s time interval

begin mathsize 16px style f o r c e space equals space r a t e space o f space c h a n g e space o f space m o m e n t u m rightwards arrow space f o r c e space equals space n u m b e r space o f space p h o t o n s space p e r space s e c o n d space space cross times space m o m e n t u m space o f space e a c h space p h o t o n end style

begin mathsize 16px style F space equals space left parenthesis 1.35 space cross times space 10 to the power of 16 right parenthesis space cross times space left parenthesis 1.24 space cross times space 10 to the power of negative 27 end exponent right parenthesis space equals space 1.67 space cross times space 10 to the power of negative 11 end exponent space N end style

 Step 6: Calculate the pressure

p r e s s u r e space equals space fraction numerator size 16px f size 16px o size 16px r size 16px c size 16px e over denominator size 16px a size 16px r size 16px e size 16px a end fraction size 16px space size 16px equals size 16px space fraction numerator size 16px 1 size 16px. size 16px 67 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px minus size 16px 11 end exponent over denominator size 16px 8 size 16px. size 16px 0 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px minus size 16px 6 end exponent end fraction size 16px space size 16px equals size 16px space size 16px 2 size 16px. size 16px 1 size 16px space size 16px cross times size 16px space size 16px 10 to the power of size 16px minus size 16px 6 end exponent size 16px space size 16px P size 16px a

Exam Tip

Make sure you learn the definition for a photon: discrete quantity / packet / quantum of electromagnetic energy are all acceptable definitions

The values of Planck’s constant and the speed of light will always be given to you in an exam, however, it helps to memorise them to speed up calculation questions!

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