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The photon model (2013)

Page history last edited by Joe Redish 2 years, 7 months ago

Working Content > Three models of light

 

Prerequisites:

 

Perhaps the strangest of all the models of light is the photon model. In Newton's 17th century "colored particle" or ray model, light consists of little particles moving very fast. Their paths form "light rays". In Huygens' model, developed at about the same time as Newton's, light was an oscillating wave -- Huygen's conjectured the vibration of some material filling all space -- an "ether". In the 19th century, Maxwell showed that what was oscillating were electric and magnetic fields -- no vibrating matter needed. Color was determined by the frequency of the oscillating fields.

 

But as the 19th century progressed and more and more was learned about matter and how it interacted with light, it became increasingly clear that none of the models really could construct a reasonable description of the emission and absorption of light by matter.

 

The solution was constructed by Einstein in the early years of the 20th century (in Einstein's "miracle year" of 1905 in which he solved 3 major problems and changed the way we look at much of physics). How this model was developed is a fascinating story -- that we will explore elsewhere. But the bottom line is that Einstein in some way combined the two models. In Einstein's picture, photons are packets of energy that can interact with matter and which are absorbed or emitted in discrete units.  One photon can be absorbed by an atom or molecule and typically results in the system being excited to a higher energy state.  It is not possible to absorb half of a photon if you only need half as much energy.  Photons occur in integer numbers. 

 

The sounds a lot like Newton's particles, but here's the catch. The amount of momentum (p) and energy (E) that a photon has can be calculated from Einstein’s photon equations. Our wave number k and angular velocity ω associated with a traveling sinusoidal wave become wave stand-ins for the particle properties of momentum and energy. All you have to do is multiply by Planck's constant, ℏ.

Of course since we know how k is related to wavelength, λ, and how ω is related to frequency, f  (see Making sense of sinusoidal waves) we can simplify this by changing to λ and f. This looks better if we use Planck's constant h instead of h-bar since then the 2π's cancel.

or to summarize

.

We could also easily express the energy in terms of the wavelength using

 

λf = c

where c is the speed of light, giving

 

E = hc/λ.

 

Therefore, the amount of energy of a photon is directly linked to its wavelength. Longer wavelength photons have lower energy than shorter wavelength photons.

 

In this equation, h is Planck's constant, h = 6.626 x 10-34 J s, and c is the speed of light, c = 3.0 x 108 m/s. This is not particularly convenient for atomic and molecular systems since we typically don't use Joules for an atom (though we do for moles of them). When thinking about single atoms and molecules interacting with light a more convenient way to express the constants is

 

hc = 1240 eV nm

 

For most purposes, a convenient mnemonic is "hc = 1234 eV nm". That's off by less than 1% and easily allows you to remember how many places there are. This is a convenient form for working with atomic and molecular systems.

 

Using this information, we can determine how many photons there are in a light source such as a laser. Lasers emit a stream of photons (or a beam of light). The amount of light emitted is typically given in Watts (where 1 Watt equals 1 Joule per second). A classroom laser pointer is often red in color, and has a wavelength of 650 nm, and a power of 3 mW. The amount of energy in a single photon at this wavelength is just E = 6.626 x 10-34 J s * 2.998 x 108 m/s / 650 x 10-9 m which is 3 x 10-19 J per photon. Since the laser has 3 mW of power, this means that there are 3 x 10-3 J / s / 3 x 10-19 J/photon = 1016 photons per second coming out of that laser. Not bad for a small hand held device!

 

Although up to now it seems like photons are just particles, you should be troubled by the fact that your particles have wave properties. This is true of everything at the quantum level. Things are even worse for photons, since we will show that a single photon can interfere with itself (a wave property), photons can get "entangled" (a quantum property) and know about each others' states even when they are far apart, and the "number" of photons in a system might not be a well defined property. While you wont need most of the quantum weirdness associated with photons for most biological applications (where all that is involved is energy conservation), the quantum properties of photons are looking extremely promising as ways to probe biological systems. Look forward to some really spectacularly interesting uses of photons in biology in the not too distant future!

 

Follow-ons:

 

 

  Workout: The photon model

 

Karen Carleton and Joe Redish

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