1. Using the wave equation and the energy equation derive an equation that expresses a photon's energy in terms of its wavelength. What proportionality exists between energy and wavelength?

## Step 1: Identify the relevant equations### The wave equation for a photon is given by , where is the speed of light, is the wavelength, and is the frequency. The energy equation for a photon is , where is the energy and is Planck's constant.## Step 2: Express frequency in terms of wavelength### From the wave equation, solve for frequency: .## Step 3: Substitute frequency into the energy equation### Substitute into to get .## Step 4: Simplify the energy equation### The final equation expressing a photon's energy in terms of its wavelength is .## Step 5: Determine the proportionality### The energy is inversely proportional to the wavelength , as indicated by the equation .