What is the Planck Length?

(Note: some of the reasoning presented here is paraphrased from Saslow WM (1998). "A physical interpretation of the Planck length." Eur J Phys. 19:313.)

What is the Planck length? German physicist and Nobel Laureate Max Planck (1858-1947) has any number of scientific feathers in his cap, but is perhaps most noted as a founder of quantum mechanics and the guy who has both a length and a constant named after him. While we're interested in the Planck length here, it's worth taking a moment to mention Planck's constant because it appears again in the Planck length.

Briefly, light is corpuscular, meaning that light as we know it is actually made of individual quantized bits called photons. Different wavelengths of light are perhaps the most obvious place to introduce energies, as most people are familiar with the idea that an ultraviolet light (invisible) has greater energy than the visible spectrum that our eyes detect. The spectrum of visible light is 400-700 nanometers (wavelength), and given that ultraviolet light is more energetic, one might be tempted to guess that it lies in the higher range of numbers relative to what we see, i.e. greater than 700. This, however, is incorrect.

Take the speed of light: if you were to derive the relevant characteristics of a particular type of light that result in a constant speed of light for a given medium, you'd be forced to the wavelength and the frequency, likely the only two things you could think of. This turns out to be correct. The product of the wavelength and frequency give the speed of light, which makes good sense to a quick analysis. Wavelength and frequency are forced to be inversely proportional, as increasing the wavelength must necessarily decrease the frequency, and vice versa. This implies that the speed of light is constant.

wavelength (lambda) * frequency (f) = the speed of light (c)

But how does this relate to the energy of light? The key lies in realizing that shorter wavelength gives a higher frequency. This can be thought of as a greater number of impinging photons and, therefore, higher energy. Thus, ultraviolet should be placed below 400 nanometers, not above 700. But how much higher energy does the frequency give us? Another quick derivation might say that the energy, by our reasoning, is equal to the frequency.

energy (e) = frequency (f)

But a rudimentary analysis might say that simply knowing how many of something we have does not tell the whole story. If light is quantized, one needs to know the energy of a quanta to know the actual energy. This turns out to be Planck's constant.

energy (e) = Planck's constant (h) * frequency (f)

The origins of Planck's constant go back to a phenomenon called black-body radiation. Basically, thermally radiated energy is just another form of quantized light, and so one might speculate that the temperature of a hot object is proportional to the amount of heat (and thus light) emitted. The object is called a "black-body" for the obvious reason that the light emitted should not include light reflected from the surface. Hotter things give off more radiation, and also produce distinct colors based on their temperature, e.g. "white" hot or whatever.

So now that we have a rudimentary idea of the Planck constant, what of the length? We've already identified the relationship between energy and frequency, so let's begin there. Again:

E = h f

Equivalently, the energy can be rewritten in terms of the wavelength for reasons already mentioned:

E = h c / lambda

But now what? What else do we know about energy or light? Perhaps this looks familiar:

E = mass (m) * c^2

Einstein's famous e=mc^2 is probably the most well-known equation in all of science. To use it, we could say that photons, by virtue of their energy, carry a certain amount of mass given that these two things are equivalent under Einstein's equation.

m = h / c * lambda

We know that light is affected by a gravitational field, because this was how general relativity was "proved" by observations done during an eclipse in 1919. But what is the relationship between light's deflection and what we've derived so far? From Newtonian gravity:

gravitational potential = -G (gravitational constant) * mass (m) / radius (r)

And thus:

gravitational potential = -G * h / c * lambda * r

Here's where it gets interesting. Imagine a situation in which an object's own mass perturbs the space within which it resides. If gravity is really a distortion of space a la Einstein, massive objects will interact with themselves by their very presence in space. For particles, this must occur at a maximum distance of the particle's own wavelength (r=lambda):

gravity = G * h/ c * lambda^2

There is a concept in physics called a Schwarzchild radius, perhaps most familiar (if unnamed) in discussions on black holes. Basically, this radius defines the event horizon, the point beyond which not even light can escape the distortion in space produced by a black hole. Roughly, the Schwarzchild radius defines the volume of space inside of which a point mass significantly interacts with and thus distorts space.

Schwarzchild radius (Sr) = 2 * G * m / c^2

For individual particles like photons, this radius could come to define a kind of unit length, if you will. When the force of gravity (or equivalently the distortion of space) is sufficiently high to capture light, i.e. G ~ c^2, lengths smaller take on a curious meaning in terms of measurement or physical interaction with the world outside.

G * h/ c * lambda^2 ~ c^2

Or, equivalently (roughly):

wavlength (lambda) = ( G * h / c^3)^1/2

As my stats professor would say, et voila! This is the Planck length, and now has some insightful physical interpretation, I think. Of course, my derivations here are rough and sloppy, but those are the most interesting ones, I think. The correct relationships are nevertheless preserved, and the physical meaning of the Planck length comes out strongly. When you hear that the Planck length is something like the smallest physical length that has any meaningful interpretation, you'll now understand why.

Thanks to Saslow from the Department of Physics at A& M for his paper.