Quantum random numbers from a laser -- simplest setup?
An alternative way to generate random numbers, that truly is quantum, and also quite easy: put a small radioactive source near a Geiger counter. Radioactive decay is a truly random event in the quantum sense, and is basically not subject to thermal noise at all.
For maximum visual impact, replace the Geiger counter with a cloud chamber. That way you can literally see the consequences of quantumly-random events. You could make random numbers from it using a web cam and some basic image processing.
If I understood correctly, what you are trying to build is a hardware based random number generator, where you want to use some quantum mechanics-based mechanism to supply the randomness.
I'm no experimentalist, thus, take my comments with a grain of salt.
Your suggestion is to use Schottky noise from a illuminated photodiode. I believe that it's a pretty safe setup (detector + light source), but your problem is that you would probably need a very dim light, and consequently a properly sealed case, in order to not completely saturate the photodiode, even if it operates at high sampling frequencies.
Even with a high sampling frequency (~ 1GHz+), I think that you would need a dim light source because of the following:
Even if your photodiode just measure optical photons, an typical optical photon has energy between 1.5 eV and 4 eV ~= 2.4-6.4 x 10^{-19} J
If you have a $\mu W$ laser, supposing that the laser is completely covered by the detector, there would be still, on the best case scenario, about $0.15 \times 10^{19-6} = 1.5 \times 10^{12}$ photons per second, so, in order to see them individually, you would need something with a THz or more sampling rate, which probably is quite expensive.
So, either you get a very low power laser, i.e. $nW$ or less, or you get a way to lower the power of a more powerful laser. One way to do that is to lower the output power of the laser and also to get a divergent lens, so not all the laser's light is deposited on the detector. If you can get a factor of 1000 lower, you might have the chance to measure with a GHz sampler.
Also, you might have to cool down your whole system in order to get the shot noise to be bigger than the thermal noise. I'm not sure if it would be feasible in room temperature, you might have to purchase a bit of liquid nitrogen which require a bit of care (proper gloves, bottle and case for it).
I would suggest that if you don't have to create a Quantum RNG, just get a thermal noise measurement as your RNG. It's probably way easier than that.
In any case, I wish you luck.
Update:
The THz estimate was based suposing that the whole detector acts as a single detection block. If it acts as an array of detectors, as would be the case using a digital camera sensor, there would be an $O(10^6)$ pixels availables in today's cameras, and this should lower the sampling frequency requirement by an equal factor.
This estimate would give a $\approx $ 1 MHz for the sampling frequency necessary to get individual photons with a 1 $\mu W$ laser, which is feasible to find in a high speed camera.
Again, using a 1 nW laser with a 1 Mpx array, the required sampling frequency would be 1 $kHz$, which is, in turn, very easy to find. Combining nW laser with Mpixel camera probably should be the easiest way to get an quantum NRG
To add to Nathaniel's Answer because (1) it is a good answer and (2) I get nervous recommending radioactive materials handling to anybody I don't know: I would really think about the cloud chamber idea, especially since you're a software guy with a math background. It would need to be inside a darkened container, but you could run a webcam to show what is going on. You can run your cloud chamber satisfactorily on either of the following sources:
Uranium glass. This is a low level source, the uranium is in glass so you can't ingest any and it's readily available as uranium glass marbles. See my answer here for possible suppliers. There is a youtube clip of a homemade cloudchamber with uranium glass in it here
Alternatively, you can make your cloud chamber big - you'll just need to have it running a bit longer before your demonstration - and you'll get quite a show from cosmic ray muon trails. You don't need any radioactive source at all. The big cloud chamber would work better with the uranium glass too. See Sheehy, S. L. "How to make a cloud chamber", produced by the Cockroft Institute at University of Oxford (google Suzy Sheehy) for some pretty good instructions about this.
Lastly, I refer you yet again to my answer here for discussion of safety measures for uranium glass handling