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In this lecture from MIT, the professor defines the intensity of a wave as the square of the amplitude of the wave. But, at the same time, the professor defines the intensity of light as the number of photons per second. Are these definitions equivalent?

I know that light has both properties of a particle and properties of a wave, but the two different definitions of intensity have different units, which seems confusing to me.

Additionally, when I searched up intensity on Wikipedia, I was unable to find a definition of intensity that matched the definition provided in the lecture. Some of the definitions of intensity that I did find on Wikipedia were "power per unit area," "power per unit solid angle," and "luminous flux per unit solid angle."

Thus, my question is: what do the terms "intensity," "intensity of light," and "intensity of a wave," mean? What are their definitions?

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    $\begingroup$ The critical thing to understand is that thee are two distinct but equally correct ways to understand EM radiation. The classical theory involves waves in fields and the quantum theory involves quanta of excitation in the fields (photons). The two definitions you've been give are equivalent, though the details of that equivalence have been omitted. $\endgroup$ – dmckee Jul 14 '18 at 21:31
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For a collimated light beam, the most relevant intensity measure is the irradiance, i.e. how much energy passes through a unit cross-sectional area in unit time (i.e. power per unit area), and for a plane wave with electric-field amplitude $E_0$, the irradiance is given by $$ I = \frac{\epsilon_0c}{2}E_0^2. $$ The (average) number of photons that pass through that area per unit time, $R=dN/(dt\:dA)$, is then obtained from the irradiance via $$ R = \frac{I}{h\nu}, $$ i.e. by dividing the energy flux by the energy of each photon as given by the Planck relation $E=h\nu$ from the light's frequency $\nu$.

As the Wikipedia page for irradiance explains in detail, there exists a huge range of radiometric measures of light intensity, depending on whether you care about the angle of emission or the spectral distribution over different frequencies or wavelengths, and so on. From these, the irradiance is the most natural measure, and (once you put in the suitable constants) all three versions of it are equivalent.

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  • $\begingroup$ The professor also defined the intensity of a wave as the square of the amplitude of the wave, but this definition is not listed here: en.wikipedia.org/wiki/Irradiance#SI_radiometry_units How does the "amplitude" definition of intensity relate to other definitions of intensity? $\endgroup$ – user62238 Jul 14 '18 at 16:00
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    $\begingroup$ It is listed. As already noted in this answer (as the very first point, though it does require you to actually read the answer), the irradiance is the square of the amplitude of the wave, once you include the correct proportionality constant. $\endgroup$ – Emilio Pisanty Jul 14 '18 at 17:40
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    $\begingroup$ And if you don't care about getting it into SI units (say because you want to talk about big picture behaviors), you can ignore the constant. $\endgroup$ – dmckee Jul 14 '18 at 21:24
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The term "intensity" has many meanings, and can be misleading depending on who interprets the word. Intensity here probably is intended to mean energy intensity- as in how luminous/energetic that light is. I feel like when your professor says "number of photons per second", they are referring to the frequency of the light wave. Professor was perhaps referring to flux of an EM wave? Its flux can be thought of as the amount of something that passes through or onto something. Like the other answer says, light is a difficult thing to describe, and there are many different properties to measure.

The *energy of a typical wave (like a real ocean wave) is given by the square of its amplitude. But in quantum mechanics however, the probability density of a wave function is given by the amplitudes square. A quantum wave function is NOT the same as a typical wave as we know it in the macroscopic world, though there are similarities mathematically. A wave function is a probability distribution. Here, intensity means frequency of observing the particle in a particular place (probability of finding the particle in a particular place.)

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    $\begingroup$ "I feel like when your professor says "number of photons per second", they are referring to the frequency of the light wave." This is not correct, in the photon picture you count photons and that is what he means. $\endgroup$ – dmckee Jul 14 '18 at 21:26

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