The photon model

In a nutshell

A photon is a small packet of electromagnetic energy. The energy of a photon depends on the frequency or wavelength of the electromagnetic radiation.

Equations

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Constants

Variables

Photons

Max Planck discovered that energy only existed in discrete packets, having a particulate nature, and Einstein called these small packets of energy photons. 

Research in quantum physics has helped us to understand that we can use different models to describe nature of electromagnetic radiation. For example we use the photon model to explain its interaction with matter, and the wave model to explain its propagation through space. The energy of a photon is described as:

​$$E= hf$$

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where $$h$$ is Planck's constant, and $$f$$​ is the frequency of the electromagnetic radiation. 

This can be combined with the wave equation $$c = f\lambda$$, to express the energy of a photon in terms of its speed $$c$$, and wavelength $$\lambda$$:

$$E = \dfrac{hc}{\lambda}$$

Example

​A laser emits green light with a wavelength of $$543\,nm$$​. Calculate the energy of each green photon emitted.

First, state the variables:

$$\lambda= 543\, nm\newline \\[0.1in] c = 3 \times 10^8 ms^{-1}$$

Then state equation:

​$$E = \dfrac{hc}{\lambda}$$

Substitute in and solve:

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$$E = \dfrac{6.63 \times 10^{-34}\times 3\times 10^8}{543 \times 10^{-9}}\newline \\[0.1in] E= 3.66×10^{−19} J$$

The energy of each photon of green light is $$\underline{3.66×10^{−19} \,J}$$​

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Quantum units​​

In the quantum world, the Joule is a very large unit, and is therefore not very useful. It is therefore necessary to use an alternative unit, the electron volt. 

The energy of $$1\,eV$$ is described as the energy transferred when an electron moves through a potential difference (p.d.) of $$1\,V$$. The work done on an electron equals p.d. multiplied by the charge of an electron $$(W = VQ = Ve).$$ Therefore the work done is:

$$W = 1V × 1.60 × 10^{−19}\,C\newline \\[0.05in] = 1.60 × 10^{−19}\, J$$

From this, it is concluded that $$1\, eV = 1.60 \times 10^{-19} J$$

The number of photons emitted by a laser can be calculated using the radiant power as shown:

​$$no. \, of \, photons = \dfrac{P}{E}$$​​

LEDs and Planck's constant​

Planck's constant can be determined by conducting an experiment using light emitting diodes (LEDs). LEDs convert electrical energy to light energy, emitting photons when the p.d. is above the LED's critical value.  

A voltmeter can measure the p.d., and if the wavelength of the light is known, Planck's constant can be determined. When the p.d. across the LED reaches the critical value, hence the LED first lights up, the work done on an electron is similar to the energy of emitted photons. In this case the two values are assumed to be equal:

$$Ve = \dfrac{hc}{\lambda}$$

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This data can be used to obtain a single value for $$h$$, but to obtain a more accurate value, the experiment should be repeated using different wavelength LEDs. A graph of $$V$$ against $$\dfrac{1}{h}$$ should then be plotted, where the gradient would be $$\dfrac{hc}{e}$$, as $$e$$ and $$c$$ are constants, a value for $$h$$ can be calculated.

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