Re: In the energy level diagram of lasers the energies are marked in +ve value

Hello Yagna,

Thanks for your question! First, let's establish you are right-the most common definition of the energy of electrons in an atom or molecule is negative. The reason for this is as follows. Imagine the entire universe were empty except for a single electron. It would make sense to define that such an electron has an energy of 0, as there are no particles pulling on this electron. Now, imagine a proton exists on the other end of the universe. The distance between the two would be so large that the electric field [1] would be negligible, and the same would be true for the for the resulting energy. Now, imagine the electron moves closer and closer to the proton. The electric field strength increases, and the electron gains velocity. This increases the kinetic energy [2] of the particle. Now, this kinetic energy must come from somewhere, and in fact it is coming from the electric field between the proton and the electron. You might have observed that this is quite similar to a marble rolling down from the edge of a bowl, and in fact it is.

An electron spinning around a proton is called a hydrogen atom [3]. Most hydrogen atoms have their electron in the lowest-possible orbit. Coming in from an infinite distance, the energy gained is 13.6 eV. This "eV" thing is a rather convenient unit of energy: it represents the amount of energy an electron gains when it is accelerated by 1 V, which is about the potential of an ordinary battery. We have just seen that setting the energy at 0 at an infinite distance is a good idea, so 0 eV -13.6 eV we have gained, gives an energy of the electron of -13.6 eV. The next-higher orbit would now have an energy of -3.4 eV, etc. We can now compute the energy gained when the electron goes from -13.6 eV to -3.4 eV as 10.2 eV. This is indeed a possibility; the resulting spectral line is called a Lyman-alpha line [4], and this is an ultra-violet line.

However, let's imagine that I'm a stubborn guy, and insist on putting the energy of an electron in free space at +57eV, because 57 is my lucky number. In this case, the electron will still gain 13.6 eV before it comes into the lowest orbit, and is now at an energy of 44.4 eV. The next orbit is at 54.6 eV. The difference (which gives the energy of the photon) is again 10.2 eV! Hence, we can add a constant energy to the system with impunity.

Another way of looking at it is the analogy of the marble in the bowl. At the bottom of the bowl, the marble (our electron) has less energy than at the edge. This is certain. Now, with respect to the floor, the energy of the marble is quite different when the bowl is at the floor, at my table or at my kitchen cupboard. However, this doesn't matter for the marble in the bowl: if you release it, it will still move around and come to rest at the bottom of the bowl in the same way, no matter where you put it[5].

Fields with this property are called conservative [6], and two very important fields share this property: the gravity field and the electric field. It is common in fields that have result from a force that decays with one over the square of the distance [7].

As a personal anecdote, during my PhD research, I happen to have simulated the behavior of excited hydrogen particles that exist in a hot, gas-like state called a plasma. I used to set the ground-state energy of the ground-state particles to 0, while the excited states got a positive energy, and the free electron/ion situation got an energy of +13.6 eV. My student, however, liked to put the energy of the ions in his simulation at -13.6 for the ground state, and 0 for the free electron. And yes, we did get the same answer :). One cannot say that one approach is better-my approach has the advantage that the "normal state" of hydrogen has energy zero, whereas my student's approach is (perhaps) formally more correct.

As for your energy diagram, the author probably decided it was most convenient to let the scale start at the ground state with energy 0. As only energy differences rather than energies matter, the eventual answer will not differ.

Hope this answers your question!

Regards,

Bart Broks

1. http://en.wikipedia.org/wiki/Electric_field
2. http://en.wikipedia.org/wiki/Kinetic_energy
3. http://en.wikipedia.org/wiki/Hydrogen_atom
4. http://en.wikipedia.org/wiki/Lyman_series
5. A pedant might note that the Earth's gravity is slightly different at one meter above the floor than at the floor. This is besides the point, and not the case in the electron/proton example.
6. http://en.wikipedia.org/wiki/Conservative_field
7. There are a few other requirements that only mathematicians care about.