Let's do an actual example with green light ~550nm wavelength = 5.5x10^-7 meters. hc ~ 2x10^-25 Joule meters, so the energy of a green quantum: E = hc/lambda = 2x10^-25/5.5x10^-7 Joules ~ 3.64x10^-19 Joules. Planck length ~ 1.62x10^-35 meters. So the wavelength of green light is 5.5x10^-7/1.62x10^-35 ~ 3.4x10^28 Planck lengths. So the total energy (using these units) is 3.4x10^28 * 3.64x10^-19 ~ 1.24x10^10 Joules. 1 kilowatt-hour ~ 3.6x10^6 Joules, so the total energy in kWh is 1.24x10^10/3.6x10^6 ~ 3.44x10^3 kWh ~ 3.44 MWh (that's a pretty healthy green laser!) The whole point of this exercise is to show that while Shannon wants to put more "information bits" into shorter wavelengths, Planck tells us that there are fewer quanta per bit at shorter wavelengths. There's also another type of problem: as the universe expands, the "same" light gets redder (= longer wavelength = more quanta). Thus, the number of quanta isn't a property of the light itself, but of something else -- perhaps the granularity of the space where it is detected. At 10:55 AM 4/8/2014, meekerdb wrote:
On 4/8/2014 10:29 AM, Henry Baker wrote:
Measure length as a multiple of Planck lengths (or some other convenient standard length), so lambda becomes unitless.
It also makes a huge difference in the number you get for total energy. What are you going to do about the h*c term which also has units of energy*length?
Brent
For this purpose, I'm not interested in the actual value, but merely the fact that it is constant.
At 10:04 AM 4/8/2014, meekerdb wrote:
On 4/8/2014 3:57 AM, Henry Baker wrote:
A single photon of an electromagnetic wave of wavelength lambda has energy E = h*c/lambda. Right.
The energy of a complete wave is computed by multiplying this equation by the wavelength to get:
Total energy = E*lambda = h*c = constant. I'm not sure what you mean by "a complete wave". EM energy comes in discrete photons, so you get the energy of, for example, a radio broadcast by multiplying the above value of E by the number of photons N, which can be any integer and doesn't depend on the wavelength. Of course in practice you do it the other way around because it's easier to measure the broadcast energy, from which you calculate the number of photons as N=(broadcast energy)/E. Your formula doesn't even have the right units; it has (total energy)=energy*length.