do the wave equations need to be modified for obstacles?

San K

do the wave equations need to be modified for obstacles?

some people have wondered if waves were to exist, which medium would they be travelling in? are they causing ripples in space-time?

for example waves (disturbance) on a ocean/pond travel in water. now if we place an obstacle we can physically/visually see the waves recoil etc and give rise to a new interference pattern that we can predict/calculate

Question:

Refer: a double slit, single particle (single particles sent one by one) experiment where a small obstacle is placed after the slits

to predict the interference pattern (and we are not, for the moment, debating if there are real waves or not, we are simply taking about the mathematics) do the wave equations need to be adjust for the obstacle to accurately predict the new (distorted) interference pattern (that forms on the screen)?

or

are we able to accurately predict the new (distorted) interference pattern in case of obstacles via modification/tweaking of the wave equations?


Note: there are two kinds of waves here

1. de broglie waves (or matter waves) ...these are hypothesized to exist

2. Probability waves (these are simply mathematical) and don't really exist

EmpaDoc

I see you are struggling to understand what all this wave business means.

First let me say that I don't understand why you want to postulate two kinds of wave. Quantum mechanics knows only one kind of wave, namely, the wavefunction of a particle (or of a system of particles). Call it matter wave, probability wave... it doesn't matter. When you are dividing it into two, you are probably thinking about the interpretation of the wavefunction which is a whole different subject (with many threads on physicsforums devoted to it!). You can say that physicists still don't know what a wavefunction "really" is since we still don't know how to interpret it in terms that we ourselves can understand. On the other hand, we know wavefunctions really well in the sense that we can use them to calculate the outcomes of experiments, and as you know quantum mechanics has been tested to astonishing accuracy in an enormous variety of settings in the past (circa) hundred years, and never failed.

Now for your question. Yes, physicists know how to deal with wave functions in the presence of obstacles. We think of obstacles in terms of the potential that are associated with them - which in turn gives rise to a force between the particle and the obstacle. It could be the electrostatic attraction between an electron or a proton, or anything else. There is a well-controlled prescription for how to include this into the Schrodinger equation, which is then solved. So I could straightforwardly predict the outcome of a double-slit experiment where you place anything you choose between the double slit and the detectors. (In many situations, that would simply effect in a changed or distorted interference pattern - unless your "obstacle" is connected to a macroscopic measurement apparatus, which is a whole different business!)

Hope I did not confuse you even more now. ;)

San K

EmpaDoc thanks for the information regarding solution in schrondinger for obstacles. Very helpful.

Your post is clear, no confusion.

question: are the obstacles considered simply absorptive or is there a "recoil" as well? I.e. Do the waves have a "bounce back" from the obstacle in the equations? any papers on this

EmpaDoc

Well, that depends on what the obstacle is. In any everyday situation, the potential will ultimately be due to the electromagnetic interaction between the electron (if we are considering an electron) and the electrons and protons in the material. It could be a conductor or a semiconductor or whatever, and they would let the matter wave penetrate into it at various rates (meaning that there is a specific probability for the electron to be found inside the object after some time if you do that measurement). Ultimately you would get some reflection and some penetration. That penetration would be quite small in many cases, I believe. Of course, in a realistic calculation, you model neither the screen nor the obstacle as a number of single atoms - because the calculation cannot be done in practice - but just as a reflecting barrier.

Papers? Not sure. At this stage, textbooks or pop science books might be more helpful, if you excuse me for sounding a bit patronizing...