Q: Does the electron in an atom make a jump from one orbit to the other without traveling in between? Is that description according to Bohr's theory correct?

A: Firstly, I would like to outline the Bohr theory: it's pretty much a model of an atom imagining it to be like the solar system. Indeed, both the Coulomb force and the Gravity force have inverse square dependence. But the difference is in the energies involved: energy is quantised in units of h, the Plank constant (since this fact was discovered by Plank in 1900), which is a very small quantity. Since the solar system is very large, h does now play a very important role there. However, atoms are very small, and so we cannot go very far treating atoms exactly like the solar system.

 There is one more difference, which was a great difficulty for the Rutherford (solar-system) model of the atom: electron are moving charges rotating around the nucleus, and a rotating (or in any sense accelerating) charge emits light continuously, thus losing energy (which the light carries away). The electron would collapse into the nucleus. Also, we don't see a continuous emission of light from atoms. And the light which IS emitted is of some very particular frequencies... so particular that we can recognize an element from it's "signature" of frequencies emitted.

 What Bohr did was reconcile Plank's hypothesis with Rutherford's theory... he said that electrons only move in certain orbits around the nucleus, and even though they still rotate, they don't emit light. They do so only when they change orbits... go from one nearer the nucleus to one further away. That changes the energy of the electron (one nearer the nucleus has less energy), and this difference in energy is emitted. Since the orbits and their energies are different for different elements, the emitted light has different energies, and thus different elements have different "spectra", or "signatures".

 The Bohr theory was puzzling in some respects: why don't electrons emit light even though they rotate? Why are only certain orbits allowed? Can an electron be in the space between the orbit while it is jumping from one to the other? (Some of our textbooks answered this with a confident "no". I don't think Bohr said so, he just said the transitions cannot be treated according to classical laws.)

 The "real" theory of the atom is the quantum mechanical theory, based on Schroedinger's equation. If you have read a little about QM, you know that all particles are treated as "waves". That is a vague way of putting it: these "waves" are in fact probability waves, which means that where a wave is strong, or have a high (crest), the particle is more likely to be found there. We cannot state where exactly the particle is: we can only give the probabilities for it to be found at the place where we are doing the measuring.

 A better description of QM in this context (Atoms) might be in the concept of "states". For example, if I measure the energy of a particle which I put in a box, it will only have certain values . Energy is almost always discretised, since (for the physics students among us), the Schroedinger equation is essentially an eigenvalue equation for the energy. (Forget that if you didn't get it.) Anyway, energy, in most situations, has only discrete values, which means measuring it will only give me one of the values from that set, and no other.  For example, I will only get 0,1,2,3,4,5... (appropriate units), but not 1.5,2.3333..., 3.14159... . I may get 1 90% of the time and 0 10% of the time, but I will never get 0.9 . This means that the particle can only be found (ie, measured to be) in certain "states".

 For example, on a flight of stairs, a person may be on the 2nd stair, or the 5th stair, but he cannot be standing in between two stairs. He may be on the 1st stair 40% of the time, on the 2nd 10% of the time, on the third 5% of the time and so on. Which means I can only find him in certain "states", which I may label state one (1st stair), state two (2nd stair) and so on.

 The quantum mechanical solution of the (hydrogen) atom says that the particle can only be found in certain states (these are the Bohr orbits: it can be in the first, nearest, orbit, the second one, or so on...). However, remember this is QM: the electron does not "rotate" around the nucleus. In fact, it has a some probability of being found anywhere in the atom, and the probability is maximum at a certain distance (which can be taken as the "radius" of it's orbit). But what it does have fixed is not the position of the orbit, but it's energy: so different orbits correspond to the different energy states. In the first state, it has a certain energy, in the second, it has another energy, etc.

 An electron also can, under some conditions, be in a superposition of two states. It being in a single state means that it will always be measured to have the same energy. A superposition means that it can be measured to have two different energies, with some probability for each. Eg., a superposition of the "first step" and "third step" means that the person can be found for sometimes on the 1st step, and sometimes on the 2nd step, with certain probability.

 A superposition of states under an oscillating external field results in an oscillation term in the the probability... ie, the electron may, at a certain time, have 100% probability of being found with one energy, and a certain time later, have 100% probability of being found with the other energy. That is, when a light of a certain frequency is incident on the atom, the electron state is changed due to the light wave, and the electron is now in a superposition of two states with different energies. So, at a certain time, it has 100% probability of being found in an upper energy state. This means that the electron has absorbed energy from the incident light. Since the electron's energy is increased by a certain amount, that means it has absorbed a "photon" of that energy from the light.

 Now, for any frequency of light incident on the electron, it is thrown into a superposition of many, many energy levels (states). But, the superposition is "strong" only if the light has an energy equal to the difference between two of the energy levels . This means that the electron will most probably transition to that upper state which satisfies this condition. (A "strong" superposition means that the electron transition probability is appreciable .) This is why light of a certain frequency will only cause certain transitions. This is responsible for the "spectrum" of the electron... ie, the absorption and emission of light of certain frequencies by a particular element.

 A visual picture can be painted if we imagine the electron as an "electron cloud" (which is what it is, after all... a single electron actually can be imagined to be a fuzzy cloud, with high probability of being found where the cloud is denser, and low probability where it is lighter). In the stationary states inside the atom, the electron has different "cloud shapes", if you will, in different states. Each cloud is densest at a certain distance from the nucleus, which we can call the "radius" of the state. Under an external oscillating field, the cloud changes shape, and becomes one stationary type of shape for some time, a "middle", superposed state for some time, and then another stationary shape for some time. Thus, the radius changes, energy is absorbed or emitted, etc. (Note that a normal superposed state has a different cloud shape than either of the two superposing stationary states.)

 Thus, the textbooks are wrong, the "radius" does change continuously when the electron is making a transition... the radius doesn't radically change.