Question about the meaning of planck's formula

[positron]

in a book i was reading, it states "the energy of a wave is proportional to the square of the amplitude. and then it says that the amplitude of an oscillating light wave can only have certain given values. but isn't the energy of a wave proportional to the frequency, not amplitude? what exactly does Planck's formula E = nhv mean and what exactly is quantized?

 

[quantumdude]

positron,

in a book i was reading, it states "the energy of a wave is proportional to the square of the amplitude.

It looks like the wave and particle pictures of light are being used interchangeably somewhere along the line here, and it further looks like that is the source of confusion.

The energy of an electromagnetic wave is indeed proportional to the square of the amplitude of the wave. But remember that this wave carries many, many photons, and the greater the amplitude the more numerous the photons. So if a monochromatic (aka fixed frequency, $\nu$) wave of EM radiation carries $n$ photons each of energy $h\nu$, then turning up the amplitude increases $n$.

and then it says that the amplitude of an oscillating light wave can only have certain given values.

That's true because $n$ is a counting number, and it determines the intensity.

but isn't the energy of a wave proportional to the frequency, not amplitude?

No, that's the energy of a single photon. This is what I mean by mixing up the wave and particle pictures of EM radiation.

what exactly does Planck's formula E = nhv mean and what exactly is quantized?

Planck actually did not develop his theory for photons. In his formula it is the energy levels of the atomic oscillators that are quantized. This formula was actually an early result in research on matter waves, not photons.

 

[CarlB]

in a book i was reading, it states "the energy of a wave is proportional to the square of the amplitude.

This is a comment about classical waves. If you have two waves, and one of them has 3x the amplitude of the other at every point in space, then the larger wave has an energy 9x as large as the smaller one.

and then it says that the amplitude of an oscillating light wave can only have certain given values.

That doesn't sound right. Maybe you could quote them exactly.

but isn't the energy of a wave proportional to the frequency, not amplitude [squared]?

It's proportional to both in the same way that the volume of a cylinder is proportional to its length and is also proportional to its radius squared.

what exactly does Planck's formula E = nhv mean and what exactly is quantized?

This equation has to do with the total energy contained in a complete photon. The difference between this way of looking at a wave and the stuff you're talking about previously is that the above stuff was about the value of the wave at a given point in space. Now you're talking about the entire wave.

That is,

$$E=nh\nu = \int_{-\infty}^\infty\int_{-\infty}^\infty \int_{-\infty}^\infty \mathcal{E}(x,y,z)dx\;dy\;dz$$

where $\mathcal{E}$ is the energy density. The above is a classical definition of the total energy. In quantum mechanics, the energy density is a bit different because of operators and all that.

The comment about energy being proportional to the square of the amplitude is a comment about the energy density. That is, $\mathcal{E}$ is proportional to amplitude squared. The Plank relationship has to do with total energy.

Carl

 

[positron]

Thanks for the replies. I realized that I was confusing the wave and particle pictures of light. In the wave picture, the energy is proportional to amplitude, and in the particle picture it is related to the frequency. It makes sense to me now.