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The energy of light is given by:

$$ E = h\nu = \frac{hc}{λ} $$

which seems weird to me is that the equation has nothing to do with its amplitude. But intuitively, since the light is wave, the energy of wave should dependent on its amplitude.

So I wondered why the energy of light/photon has nothing to to with its amplitude according to the above equation?

John Rennie
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    A photon does not have amplitude. Amplitude refers to the number of photons. – safesphere Jun 12 '18 at 03:22
  • Einstein’s resolution of the photoelectric effect (that light energy is only related to frequency, not intensity) would say that there is no relevance of the intensity to the energy. – Matt Hanson Feb 16 '24 at 03:18
  • In the photoelectric effect, intensity does have an effect on the current produced, if the light frequency is above a threshold value, but if the frequency is below the threshold, then intensity has no effect . This is not what is expected if the light was behaving as a classical wave. The difference is that the intensity is the number of photons impinging on the photoelectric cell per second and individual photons do not have an intensity. – KDP Feb 16 '24 at 08:16

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Light is one way that the electromagnetic field can behave. Static electric and magnetic fields are more examples of the electromagnetic field. The electromagnetic field is a complicated beast and to describe it properly requires a grasp of special relativity and vector calculus. Classically we use Maxwell's equations, and in quantum mechanics we use a theory called quantum electrodynamics.

Light is, speaking loosely, an oscillation in the electromagnetic field that travels through space much as a water wave travels across a water surface, and when we talk about the amplitude of the light we mean the amplitude of this wave. However the electromagnetic field is quantised, and one result of this is that an electromagnetic wave can't just have any amplitude. The energy of the light wave of frequency $\nu$ has to be multiples of $h\nu$ i.e. multiples of the photon energy.

Imagine we start with a wave of zero amplitude and we want to build it up by adding energy to it. The minimum amount of energy we can add is $h\nu$, and that gives us a light wave corresponding to a single photon. Add another $h\nu$ and we get a light wave corresponding to two photons, and so on. Since the energy of a single photon is so small the sort of light we see in everyday life corresponds to huge numbers of photons. For example sunlight has an energy of around $0.$ to $1$ kilowatt per square metre depending on where in the world you are, and if we take an average wavelength of around $500$nm that's of the order of $10^{21}$ photons per square metre per second.

So the answer to your question is that a light wave is a state built up from many ($10^{21}$ !!) photon states, and the state that we call a light wave has an amplitude that is a separate concept to the photon energy.

But be cautious about imagining a light wave as a hail of photons. A light wave is a state of the quantised electromagnetic field that can be expressed (approximately) as a combination of many photon states but it isn't simply a collection of photons. It is simpler to regard the photon as the unit of energy exchange. Whenever the electromagnetic wave exchanges energy with anything else it does so in multiples of $h\nu$ i.e. in integral numbers of photons.

If you're interested I discuss photons in more detail in my answer to Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$?

John Rennie
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Amlitude enters when the wave theory of light is considerd. For instance light of greater amplitude would have many of these photons.

SAKhan
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Because the equation you are considering is provenient from a quantum interpretation of light. On this ground, light is a flux of massless particles traveling in space. Each of those particles have energy $E= h\nu$.

On the other hand, when you consider light as a wave, the energy is defined by the Poyinting Vector $\mathbf{\vec S= \vec E \times \vec H}$