Frequency of an EM wave in Classical and Quantum Physics

[Hak]

In Classical Physics, when a charged particle oscillates, it emits an electromagnetic wave, and the frequency of the wave depends on the frequency with which the particle oscillates.
But in Quantum Physics, when an excited atom emits a photon, the energy of the photon depends on the amplitude of the quantum jumps that the emitting electron makes (if it jumps one level, the photon will have a certain energy; if it jumps two, a higher energy, and so on). So the frequency of the electromagnetic wave corresponding to the photon will depend on the amplitude of the quantum jumps made by the electron.

I do not understand why these two cases are so different. In analogy with the classical case, shouldn't the frequency of the wave emitted by the atom depend on the frequency with which the electron makes quantum jumps? Or is there a quantum explanation of the classical case that I do not understand?

I know that one cannot use classical physics to explain quantum phenomena, but it seems strange to me that there is this asymmetry in the two cases. Sorry in advance if the question is dumb; I am approaching Quantum Physics because it is a subject I am so passionate about, but I have a totally different background... Thanks if you can give me some clarity!

 

[Frabjous]

Check out this thread.
https://www.physicsforums.com/threa...eration-relate-to-the-emitted-photon.1001351/

 

[Mister T]

So the frequency of the electromagnetic wave corresponding to the photon will depend on the amplitude of the quantum jumps made by the electron.

You need large numbers of photons for the properties of the electromagnetic wave to appear.

 

[PeterDonis]

I do not understand why these two cases are so different.

One case is a free electron oscillating. The other case is a bound electron changing energy levels. These are very different states of an electron, so I do not understand why you would not expect them to be so different.

is there a quantum explanation of the classical case

Yes; more precisely, there is a quantum explanation of a key difference in the two cases. In quantum mechanics, the energy spectrum of a free particle is continuous, but the energy spectrum of a bound particle is discrete. That means their interactions with the electromagnetic field will be very different.

 

[vanhees71]

In Classical Physics, when a charged particle oscillates, it emits an electromagnetic wave, and the frequency of the wave depends on the frequency with which the particle oscillates.
But in Quantum Physics, when an excited atom emits a photon, the energy of the photon depends on the amplitude of the quantum jumps that the emitting electron makes (if it jumps one level, the photon will have a certain energy; if it jumps two, a higher energy, and so on). So the frequency of the electromagnetic wave corresponding to the photon will depend on the amplitude of the quantum jumps made by the electron.

This picture is not correct according to modern QT (the theory discovered in parallel by Heisenberg, Born, Jordan and Schrödinger and Dirac in 1925/1926).

The stationary states of an atom are the eigenstates of the Hamiltonian, neglecting the coupling to the quantized radiation field, i.e., taking only the Coulomb interaction between the atomic nucleus and the electrons and between the electrons (as well as their magnetic moments to get the fine and hyperfine structure right) into account.

The coupling to the quantized electromagnetic field is then taken into account by perturbation theory. This leads to spontaneous emission of photons, where the atom makes a transition of some excited to a lower state. The emitted photon has the energy given by the energy difference between these levels (with some finite width due to the life-time of the excited state according to the corresponding energy-time uncertainty relation).

I do not understand why these two cases are so different. In analogy with the classical case, shouldn't the frequency of the wave emitted by the atom depend on the frequency with which the electron makes quantum jumps? Or is there a quantum explanation of the classical case that I do not understand?

You can describe this case also quantum mechanically. It's induced emission/bremsstrahlung due to a oscillatory external force on the electron, e.g., due to an electromagnetic wave. In this case what's usually produced is a coherent state of the em. field. For classical situations, i.e., with many electrons brought, into collective oscillations (aka a classical AC) you get a coherent state of pretty high intensity, and that's very well described as a classical em. wave. So there's a clear quantum description also of classical situations.

I know that one cannot use classical physics to explain quantum phenomena, but it seems strange to me that there is this asymmetry in the two cases. Sorry in advance if the question is dumb; I am approaching Quantum Physics because it is a subject I am so passionate about, but I have a totally different background... Thanks if you can give me some clarity!

It's not dumb. It's a very valid question, how to understand the success of classical electrodynamics given the underlying quantum nature of all phenomena!

 

[WernerQH]

But in Quantum Physics, when an excited atom emits a photon, the energy of the photon depends on the amplitude of the quantum jumps that the emitting electron makes (if it jumps one level, the photon will have a certain energy; if it jumps two, a higher energy, and so on). So the frequency of the electromagnetic wave corresponding to the photon will depend on the amplitude of the quantum jumps made by the electron.

I don't know what you mean with "the amplitude of the quantum jumps" -- I can't make sense of it at all. The amplitude, both in quantum as in classical theory, is related to the intensity of the radiation, not its frequency. In quantum theory, the frequency is related to the energy difference between the initial and final states of the electron, $h\nu = E_i - E_f$.

The parallels between the quantum and classical descriptions may become clearer if you consider highly excited states of hydrogen (so-called Rydberg states). The energies are proportional $1 / n^2$, and the energy differences proportional to $1 / n^2 - 1 / (n + 1)^2 \approx 2 / n^3$. In a highly excited state ($n \gg 1$) the electron moves at a much larger average distance from the nucleus with a much lower speed. (Just like the outer planets do in the solar system.) This means longer period, and lower frequency of the emitted radiation ($\propto n^{-3}$).

In a magnetic field, electrons emit cyclotron radiation at the Larmor frequency $\omega = e B / m$, which corresponds exactly to the constant spacing between the Landau levels. For non-relativistic electrons the emitted field varies sinusoidally with that frequency, i.e. electrons step down the Landau levels one by one. But when they become relativistic, in the classical description the field no longer varies sinusoidally, but contains overtones, higher harmonics. In the quantum picture this means that there are jumps with $\Delta n \gg 1$.

The classical picture assumes that motion is continuous. But quantum theory reveals that, in the real world, motion and radiation occur discontinuously, in very small but finite steps.

 

[Hak]

Thank you very much.