user: When light enters a medium other than a vacuum it slows down. When Light exits the medium and enters back into the vacuum it speeds back up. Where does the energy come that accelerates the light?

assistant: Einstein's famous theory of relativity is based on the general equation:
```
E^2 = p^2*c^2 + m^2*c^4
(E = m*c^2 is derived from this)
```
This tells us that energy **E** squared is equal to momentum **p** squared times times the speed of light **c** squared, plus mass **m** squared times the speed of light **c** to the fourth power.

Light is massless, but it can still have momentum. So, for the energy of a photon, we can simplify the general equation to get:
```
E = p*c
```
Photons are massless particles, and so get all their energy from momentum. If they were to ever run out of momentum, they would cease to exist.

When a photon enters a medium other than a vacuum it slows down. Since energy cannot be created or destroyed, this means that momentum increases to compensate! Since the new medium is not a vacuum the photon now has other particles it can bounce off. Because the speed of light for that medium is constant, whenever it hits a particle the photon transfers some of it's momentum/energy to it. Since light is both a particle and a wave, we can observe the light's reduced energy by looking at how the wavelength of the light changes. Shorter wavelengths of light like ultraviolet light are high in energy, while longer wavelengths like radio waves have lower energy. When the photon returns to a vacuum, the speed of light increases. But remember, energy can't be created or destroyed, so the photon's momentum decreases to compensate for the increased speed.

user: Describe how light can be both a particle and a wave.

assistant: The concept of wave-particle duality was first proposed by French physicist Louis de Broglie in his PhD thesis in 1924. He suggested that just as light can behave as both a wave and a particle, particles such as electrons can also exhibit both wave-like and particle-like properties. This idea was initially met with skepticism, but it was later supported by experiments such as the double-slit experiment, which demonstrated the wave-like behavior of electrons.

Light can exhibit properties of both a particle and a wave because of its dual nature. This is known as wave-particle duality, a concept that emerged from quantum mechanics.

When light is observed as a wave, it exhibits properties such as diffraction, interference, and polarization. These characteristics are typical of waves and can be explained by the wave-like behavior of light. For example, when light passes through a narrow slit, it spreads out and interferes with itself, creating a characteristic pattern of bright and dark fringes on a screen behind the slit.

On the other hand, when light is observed as a particle, it exhibits properties such as quantization and momentum. These characteristics are typical of particles and can be explained by the particle-like behavior of light. For example, when light interacts with matter, it can transfer its energy in discrete packets called photons. The energy of each photon is directly proportional to the frequency of the light wave, which is a wave-like property.

The wave-particle duality of light can be explained by the wave-particle duality of matter, which is a fundamental concept of quantum mechanics. According to this concept, all particles, including light, exhibit both wave-like and particle-like properties. Therefore, light can behave as both a wave and a particle, depending on the experiment and observation method used.