# Can an electron jump to a higher energy level if the energy is insufficient or exceeds the $\Delta E$?

Lets say we have an atom of hydrogen. It has one electron on $E_1 = -13.6~\mathrm{ eV} ~~(E_2 = -3.4~\mathrm{eV})$ energy level. I know that if we fire a photon with 10.2 eV energy the hydrogen atom will absorb it and the electron will jump on the next energy level E2. And bellow are my questions.

• Q1: If a photon with 10.1 eV energy (insufficient to excite electron) would hit the atom of the hydrogen what would happen? Will the photon be absorbed by the atom and immediately emitted and the emitted photon (or photons?) will have the same 10.1 eV energy? Or the photon will pass through the atom or what would happen?
• Q2: Same question as the above one in this case our photon has a slightly more energy lets say it has 10.3 eV. Again what would happen? Will the atom absorb the photon and excite the electron but since the energy of the photon exceeds the required energy to excite the electron will the atom emit a photon with 0.1 eV energy or what will happen in this case?

I have done some research about it and got really confused. Some say that it needs the exact amount of energy ($\Delta E= E_2 - E_1$ in our case $\Delta E$ equals to 10.2 eV) to jump onto the higher energy level some say that it can jump if the energy exceeds the $\Delta E.$ What I really could not find is what happens with the extra amount of energy or maybe electron can be on $E_2$ energy level with slightly more/less energy.

Eventually I want to understand the concept of the reflection. How we see the objects, why they are transparent or glossy or red or whatever else. But this is out of scope of my question.

I'm not an expert though; so mark done the mistakes above if there are any.

• Interesting question... in the semiclassical model of light-electron interaction, there is a very small (but finite!) probability of an electron moving to a higher level even when the frequency of the light incident is less than the energy gap; I wonder if the same result exists in the full-quantum model of light-electron interaction. I suspect energy-time uncertainity has something to do with this. – Harsha Dec 2 '16 at 7:40

## 2 Answers

When a photon hits a boundary condition , three things can happen: a) it can scatter elastically, which means it retains its frequency but changes angle, b)it can scatter inelastically, which means it changes frequency, or c) it can be absorbed raising the energy level of an electron ( in a lattice, in a molecule, in an atom) and a different photon is emitted and phases are lost.

Q1: If a photon with 10.1 eV energy (insufficient to excite electron) would hit the atom of the hydrogen what would happen? Will the photon be absorbed by the atom and immediately emitted and the emitted photon (or photons?) will have the same 10.1 eV energy? Or the photon will pass through the atom or what would happen?

The hydrogen atom hit with a photon of energy lower than an energy level transition falls under a) or b) The photon will scatter elastically in the center of mass with the total atom and go on its way at adifferent angle, or inelastically giving kinetic energy to the whole atom and changing frequency.

Q2: Same question as the above one in this case our photon has a slightly more energy lets say it has 10.3 eV. Again what would happen? Will the atom absorb the photon and excite the electron but since the energy of the photon exceeds the required energy to excite the electron will the atom emit a photon with 0.1 eV energy or what will happen in this case?

If the extra energy of the photon is not within the energy width of the hydrogen energy level, again it will go on its way scattering elastically or inelastically in the center of mass "photon atom" . If the energy of the photon is higher than the ionization energy of the atom, the work function, the electron may be kicked off and the ion proton remain. The photoelectric effect.

One has to realized that at the quantum mechanical level it is probabilities that are important. The probability for a photon of the correct energy difference to raise the electron of an atom is very high, with the wrong energy difference. very very small.

For bulk matter interaction see this answer of mine here.

• So basically as I understood there is one thing that we know for sure that if and only if the energy of photon is equal to an energy level transition of an atom the absorption will occur. And when the energy is less or more one of 2 things can happen. And what exactly would affect photon to go with a) or b) or it is something that is not possible determine which is actually even worse. As far as I can understand it has something to do with the atom(molecule) itself? – Edward Chopuryan Dec 2 '16 at 11:42
• @EdwardChopuryan not exact but within the width of the energy state. It is a matter of boundary conditions and probabilities for the specific problem as well as the energy of the photon; there will be a calculable probability for elastic scattering, and another one for compton like scattering ( inelastic with loss of energy). What happens to a specific photon is a matter of probabilities. – anna v Dec 2 '16 at 15:55

If your photon has not enough energy to excite the electron then it will just not be absorbed and will pass by, and if you have an electron with an excess energy, it will be absorbed and a photon with the excess energy will be automatically emitted and the electron will jump to an excited state. So yeah in your case, you might have a photon with $0.1 \space \mathrm{eV}$ of energy which corresponds to a photo with a wavelength of $12.4 \space \mathrm{\mu m}$ so in the infra-red.

• It doesn't work quite that way. Energy and momentum conservation don't allow for the simple emission of a second photon with the difference energy unless the atom is coupled to other atoms, e.g. in a high pressure gas. That's why single atoms (or low pressure gases) have clean line spectra with natural line widths that are given by the coupling to the vacuum (neglecting Doppler shift). I I not aware that there is a hydrogen line with 0.1eV natural line width. – CuriousOne Nov 1 '15 at 23:46
• Hmm yeah you are right, did not thought of that. But then this is a great question, what happens with the $10.3 \space eV$ photon? – user97166 Nov 2 '15 at 1:18
• Well, even atomic hydrogen has a very small index of refraction, so there must be a tiny bit of momentum transfer to the atom (classically one would call that light pressure), which means that we do have some energy transfer, but not into the internal degrees of freedom. For the purposes of atomic physics we typically neglect that completely. It gets further complicated by the fact that there is stimulated emission and for sufficiently large photon densities we can have two-photon processes which will mix different frequencies and even lead to photo-ionization below the ionization energy. – CuriousOne Nov 2 '15 at 1:32
• Part of the reason why I didn't attempt to write an answer to this question is because I don't know how to sum up the entirety of possible effects and the different levels of explanation for them... atomic physics fills textbooks for a purpose. – CuriousOne Nov 2 '15 at 1:34
• @CuriousOne So if short then we don't exactly know what would happen with the 10.3 eV photon? Also lets say we have more than one atoms, doesn't necessarily the hydrogen. What would happen in that case if photon's energy exceeds the required energy or it's insufficient? – Edward Chopuryan Nov 2 '15 at 8:11