Optics and Refraction

A monochromatic red ray of light, enters a beaker of water , $\gamma = \gamma ' / \mu $ , $\mu $ is the refractive constant.

The wavelength should reduce, (taking $\mu$ as $1.33$ ), so the light should become green, but we see it as red light. Why?

Answer

This is an important question because it gets at the concept of color, and what that really means. Mitchell is correct that when you observe scattering from a laser beam sent through a beaker of water, you are really only seeing the photons which have left the water ($\mu\approx1.33$) and the beaker (glass $\mu\approx1.5$), traveled through the air ($\mu\approx1.00029$), entered your eye ($\mu\approx1.33$), and were absorbed by your retina. So which color do you see? And does it change if you repeat the experiment underwater or use oil ($\mu\approx1.5$) in the beaker instead?

The key to understanding why we always see red light in this experiment rather another color is due to the nature of your photoreceptors, which (like most light absorbers) are not concerned with the wavelength of the light. They instead respond to the frequency of the light, $\nu$ (or photon energy $E=h\nu$), which remains unchanged as light propagates from one (linear) medium to the next. The reason for this is that there are certain energies, determined by quantum mechanics, which are assigned to the electronic states in the absorbing material (your retina, in this case). In order to conserve energy, the photon needs to have the right energy to be absorbed, promoting the material from a low energy state to a high energy state. Wavelength has nothing to do with it. Indeed, when we say "color" in reference to light, we are often actually speaking of the frequency rather than wavelength. When we do refer to a wavelength as a color, it is always with an implicit assumed refractive index, usually $\mu=1$.