Why electrons do not emit electromagnetic radiation when they "jump" to an excited state?

Question

According to electromagnetism, accelerating charges emit electromagnetic (EM) radiation. However, according to quantum mechanics, electrons do not emit EM radiation while they "orbit" around a nucleus.

When excited electrons "jump" from a high-energy orbital to a lower-energy orbital, they release the excess energy in the form of EM radiation, or in other words, they emit a photon.

Q1) When an electron in its ground state absorbs a photon, it absorbs its energy and "jumps" to a higher energy orbital. While it transitions between these orbitals, does it emit EM radiation? If so, of what frequency? If not, what prevents it?

If an electron emits electromagnetic radiation as it transitions between orbitals, this should be the case independently of the electron ending up in a higher energy or a lower energy orbital.

Q2) If an electron absorbs a photon, the electromagnetic energy is transformed into kinetic energy. Can an electron absorb a photon only when bound to a nucleus? Is it right to think that if an electron in empty space absorbs a photon, it causes it to accelerate and this acceleration on a charge simultaneously causes the release of EM radiation (another photon) perhaps of lower frequency?

Answer 1

According to electromagnetism, accelerating charges emit electromagnetic (EM) radiation.

This is according to classical electromagnetism.

In classical electrodynamics, sources can be point particles with perfectly well-defined trajectories $x_i(t)$ and perfectly well-defined momenta $m_i \dot x_i(t)$. This is not possible in quantum electrodynamics.

However, according to quantum mechanics, electrons do not emit EM radiation while they "orbit" around a nucleus.

In quantum mechanics, electrons do not emit radiation when they are in stationary states, i.e., eigenfunctions of the Hamiltonian. This was Niels Bohr's bold proposal from more than 100 years ago.

Note also that electrons in an atom are not "orbiting" anything in the classical sense, despite the fact that we use the term "orbital" to describe a stationary state. (Do not confuse the language with the physics.)

Q1) When an electron in its ground state absorbs a photon, it absorbs its energy and "jumps" to a higher energy orbital. While it transitions between these orbitals, does it emit EM radiation?

No, it does not emit anything when it transitions from a low energy state to a higher energy state. As you wrote just above, it absorbs a photon in order to have sufficient energy to get to the higher level.

If not, what prevents it?

Conservation of energy.

If an electron emits electromagnetic radiation as it transitions between orbitals, this should be the case independently of the electron ending up in a higher energy or a lower energy orbital.

You assert that the electron "should" do something, but your assertion is unfounded.

Q2) If an electron absorbs a photon, the electromagnetic energy is transformed into kinetic energy.

To say that the electromagnetic energy is transformed into kinetic energy is potentially misleading. For example, for a hydrogenic atom, the kinetic energy of the higher energy state is actually lower! You can see this from the virial theorem for a hydrogenic atom, which shows that, for a stationary state: $$ 2\langle\hat T\rangle + \langle \hat U\rangle = 0\;,\tag{A} $$ where $\hat T = \frac{\hat {\vec p}^2}{2m}$ and $\hat U = \frac{-Ze^2}{|\hat {\vec{r}}|}$.

Of course we can also write Eq. (A) as: $$ \langle \hat H\rangle + \langle \hat T\rangle = 0 $$ or, for a hydrogenic orbital $$ E_n = -\langle \hat T\rangle\;. $$

So, for example, in the ground state of hydrogen the total energy is: $$ E_0 = -13.6 \textrm{eV} $$ and the average kinetic energy is: $$ T_0 = +13.6 \textrm{eV}\tag{B} $$ and the average potential energy is $$ U_0 = -27.2\textrm{eV} $$

And, for example in the first excited state, the total energy is $$ E_1 = -3.4 \textrm{eV} $$ and the average kinetic energy is: $$ T_1 = + 3.4 \textrm{eV}\tag{C} $$ and the average potential energy is $$ U_1 = -6.8\textrm{eV}\;. $$

By comparing Eq. (B) to Eq. (C) you can see that the kinetic energy of the first excited state is actually lower than the kinetic energy of the ground state.

Can an electron absorb a photon only when bound to a nucleus?

It does have to be bound in some way. Otherwise there is no way to satisfy conservation of both energy and momentum.

Is it right to think that if an electron in empty space absorbs a photon...

An electron in empty space can not absorb a photon for the reasons stated directly above.

Answer 2

Q1) When an electron in its ground state absorbs a photon, it absorbs its energy and "jumps" to a higher energy orbital. While it transitions between these orbitals, does it emit EM radiation? If so, of what frequency? If not, what prevents it?

First of all, it is not the electron that absorbs the photon energy, but the whole system nucleus-electrons.

The transition can be regarded in different ways. The simple verbal description that is so often used draws photon as a localized object (a mass-less point) with definite energy and momentum (and sometimes, even position), and the absorption on atoms or scattering on electrons to be a very short, or even instantaneous process. This is an extremely simplified and seriously misleading description of how our equations describe the interaction. This can be called a "buckshot theory of photons", because we have those flying things with definite momentum that get localized in all pictures showing interaction with charged particles. There is no room in this story for "EM field" or "EM radiation" with their well-known and verified properties such as frequency of oscillation in time, or polarization. It is thus just a simple story to tell which recovers some results of the theory.

However, in the best equations we have, EM radiation is described by a field that exists everywhere, and this field evolves continuously with time (at least, most of the time), including during the transition. The transition is a process in time that takes some time, during which the atom is not in "stationary" state, that is, state with well defined energy, but is in superposition of two or more such states. During such superposition, average expected current density in the atom is non-zero, and oscillates in time in a complicated way, as a sum of oscillations at frequencies $f_{mn}$ that are proportional to differences of eigenenergies $E_m,E_n$ of the atom.

Different frequency oscillations of the current are present with differing intensity, related to initial state of the atom and character of the incoming primary EM radiation. If the incoming primary EM radiation frequency is tuned to a transition between two eigenstates, current oscillation at this transition frequency becomes the strongest.

To be consistent with Maxwell's equations, there should be a secondary EM radiation due to electrons, centered at the region where the atomic electrons oscillate. When the system is in a process of absorbing the incoming radiation, the system produces EM waves of the same frequency but almost opposite phase, so the EM radiation in the original direction behind the atom is strongly suppressed (absorption). When the system is in a state of stimulated emission, it radiates with phase that is in-phase with the incoming EM radiation, and thus behind the atom, it boosts the incoming EM radiation in the original direction.

Q2) If an electron absorbs a photon, the electromagnetic energy is transformed into kinetic energy. Can an electron absorb a photon only when bound to a nucleus? Is it right to think that if an electron in empty space absorbs a photon, it causes it to accelerate and this acceleration on a charge simultaneously causes the release of EM radiation (another photon) perhaps of lower frequency?

Electron does not absorb the photon, the whole atom does. And this energy does not all turn into kinetic energy, but some of it turns into potential energy (because the whole system gets into excited state with higher expected average of potential energy).

When the electron is alone, not a part of any atom or other composite system, it can only scatter the incoming radiation, meaning that while the electron is gaining some energy (and change of momentum), it produces also outgoing EM radiation with some energy and momentum leaving. This experimentally implied fact can be explained by the assumption that electron cannot, during interaction with EM radiation, easily change its rest mass. A massive body completely absorbing another body with energy and momentum (such as a mass-less packet of radiation) would lead to increase of mass of the massive body. If increase and decrease of mass does not happen, then the interaction process must result in some energy and momentum leaving away from the electron.

Thus electron in empty space cannot, as far as we know, absorb a photon, it just does not happen. If it happened, the electron would increase its mass, but we do not observe heavier electrons resulting from scattering of EM radiation off the electrons.

But indeed, the scattering process near single electron, where radiation transfers some energy and momentum to the electron, and another outgoing radiation with somewhat different characteristics is produced by the electron, can be understood in classical theory as the electron being accelerated by the primary radiation, and thus radiating its own secondary radiation. This secondary radiation can have variety of different directions, intensity then depending on this direction in the expected way (e.g. in case of linearly polarized EM radiation, the strongest scattering will be in the plane perpendicular to electric field of the primary wave). The scattered radiation can have shifted frequency depending on direction, if the electron moves in the lab frame; this would be an example of the Doppler effect.

Answer 3

The question draws on classical physics to get intuition about quantum physics, and this is a correct thing to do.

In classical electromagnetism it is true that we ordinarily say that an accelerating charged particle emits radiation. This carries over to the quantum case when we have, for example, an oscillating electric dipole. An atom in a state which is a superposition of two energy eigenstates carries such an oscillating dipole and it emits radiation. This is the situation commonly described as a 'quantum jump'. It is not really a jump, it is a process which evolves over a timescale of the order of nanoseconds in some cases, and much longer or shorter in other cases. However if you have a photon detector with a response time faster than the time the atom takes to evolve, then the detector will 'click' at some particular time.

Now let's come to absorption. The classical situation now might be, for example, a dipole which is oscillating in some potential well and someone fires at it a carefully focused and timed pulse of radiation. If you got the focusing and the timing just right then this would result in bringing the oscillating dipole to rest. So you see, absorption of radiation can be associated with acceleration in the classical case. Similar remarks apply to the quantum case. Now you have a light field impacting on an atom, and it drives the electron into just that oscillation which is appropriate to absorb one photon of that light. Once the photon has been absorbed the electron has reached its final state, typically an energy eigenstate, and the light field has reduced energy. Typically the light is now no longer resonant with further transitions in the atom, so it won't cause further absorption. However it could cause stimulated emission.

(If the stimulated process is faster than the spontaneous decay then indeed the atom is immediately driven down to its lower state again and then back up and so on---the phenomenon called Rabi oscillation. If the spontaneous decay is faster on the other hand, whether on this transition or another, then it is the one that dominates and you don't get Rabi oscillation.)

Finally, you are quite correct that an electron in empty space cannot either emit or absorb a single photon. In the case of an atom it is not really a single electron that emits or absorbs, it is the whole atom. However in the joint wavefunction of the atomic electrons only one part changes, and for this reason we speak of a 'single electron transition'. You can also have two-electron transitions but the question was not about that.

Answer 4

Q1) When an [bounded to an atom] electron … absorbs a photon, it absorbs its energy and "jumps" to a higher energy orbital. While it transitions between these orbitals, does it emit EM radiation? If so, of what frequency? If not, what prevents it?

The exciting photon must have a minimum energy. If it has a higher energy, the electron can also move away from the atomic nucleus over several energy levels. The portion of energy, that cannot be transferred to the electron (because nature has arranged it so that the electron can only have certain distances/probabilities of residence/energy levels), is re-emitted.

Q2.1) If an electron absorbs a photon, the electromagnetic energy is transformed into kinetic energy.

The absorption of a photon by a bound electron increases its potential energy.

Q2.2) Can an electron absorb a photon only when bound to a nucleus? Is it right to think that if an electron in empty space absorbs a photon, it causes it to accelerate and this acceleration on a charge simultaneously causes the release of EM radiation (another photon) perhaps of lower frequency?

This is a question that is controversially discussed in the forum. The following example is a good argument: Since electrons can be accelerated with the help of laser beams ("Direct acceleration of electrons in a coherent, intense light field is revealed by a remarkable increase of the electron number in the MeV energy range." Amplification of Relativistic Electron Bunches by Acceleration in Laser Fields), the energy of the photons must - at least partially - be transferred to the electron.

And, yes, if the conversion of photon energy to the kinetic energy of the electron is only partial, then this causes the release of another photon of lower frequency.