Actually, there is no quantum mechanics needed, only classical electromagnetism. The calculations were first done in 1908, by Gustav Mie.
In this experiment, the gold is forming small colloidal spheres. These spheres scatter light according to Maxwell's equations, just like small spheres of water (raindrops) scatter light (making rainbows, etc.) I think that gold absorbs light, which means its refractive index is complex (has an imaginary part), while water is just 3/4 or something like that (and real, meaning no absorption). Also, the sizes are different.
The mathematical problem is, however, identical for the two cases (gold sols, rainbows). The Mie scattering solution applies to both. (In fact, understanding the colors of gold sols was the original reason Mie had for pursuing this.) Anyways, you can use this to calculate the spectrum and color for various sizes of gold spheres. It is a fairly involved calculation, I doubt you are expected to duplicate it. The results for gold spheres happen to be different colors for different sizes. If I am reading the notation in my book right, for spheres diameter dia = 0.04 micron, you get red. dia = 0.10 micron makes purple, dia = 0.14 micron makes blue.
I haven't done the calculation for gold sols, although I have done it for rainbows.
The book 'Light Scattering by Small Particles' by H. C. van de Hulst (a Dover book, so cheap) has all kinds of cool info about this kind of stuff.