I am a mathematician and I am studying string theory. For this purpose I studied quantum theory. After reading Feynman's book in which he described the double-slit experiment (Young's experiment) I was wondering if I send one electron per day or per month (even more), could I see the interference pattern?
Yes, the interference pattern will occur, although you'll have to wait a while to be able to see it. As long as the average arrival time between photons is markedly greater than the travel time from slit to detector, the actual rates don't matter - each photon interacts with the slits by itself.
This URL shows such an experiment, in which a laser beam was so attenuated that the separation between photons was in the kilometer range, while the target-detector distance was in the meter range, and an image intensifier was used to detect photon positions. After about 500,000 photons had been detected, the result was
Yes,you will see the interference pattern,time doesnt matter if the conditions are same. If you send one electron it will hit particular point on the screen,you cannot predict where it will hit,but ofcourse you can predict the probability of hitting a particular point. after many days,most of the electrons will hit the most probable regions and few hit areas of less probability and you will see the interference pattern eventually.
Imagine that the sended electron interacts with the surface electrons from the slits edges. Together they form a quantized electric field. This field is not static in the sence that the position of the incoming electron is slightly different and the surface electrons are not standing still. The incoming electrons get deflected from the surface electrons (or hits the wall or goes trough without influence if it is fare away from the edge) and this deflection is quantized.
The intensity distribution on the detector's screen shows this quantized field. So it does not matter you shot a electron per day or per month. About the somehow similar interaction between the surface electrons and photons see here.
Yes, the electron is discribed not by a path, like a macroscopic object, but by a wavefunction. And if an undisturbed electron (we better say an undisturbed wavefuction associated with an electron) goes through the slit it, just like a normal wave, interferes with itself, producing an interference pattern that will become visible if you only wait long enough.
When a time interval between photons or other particles that bombard a foil with two slit is more than milliseconds, the phenomenon of the double slit diffraction/interference will be absent.
For example, in a paper: V. Krasnoholovets, Sub microscopic description of the diffraction phenomenon, Nonlinear Optics, Quantum Optics, Vol. 41, No. 4, pp. 273 - 286 (2010); also http://arxiv.org/abs/1407.3224 the author describe a mechanism of the diffractionless of photons on one pinhole/slit in the case of a very low intensity of statistically single photons. NOTE: the author describes REAL EXPERIMENTS.