Waves - Length, Frequency and Energy

To understand the equations used in Spectroscopy and to understand the ideas involved in Schrödinger's theory, you need to know some basics about waves. Waves are commonly drawn like this: 

However, in the real world it is not always easy to see waves this way. For instance, we think of waves at the beach as looking like this:

But we get a better picture of those waves if we zoom out. From this view we can see the wave (drawn here in red). 

Sound waves can't be seen, but we know that they are a pattern of denser (higher pressure) air and less dense (lower pressure) air. We can "see" the wave the same way.

In all cases, waves involve the movement of energy in some way or another. Waves in water can carry enough energy to knock down someone wading in the surf, to swamp a boat, or to cause massive destruction in a tsunami. Sound waves can be as gentle as a whisper or forceful enough to break an eardrum. Light waves are actual pulses of pure energy which can warm you gently or slam into your skin causing irritation and cancer. 

Although all of these waves are different, they all have some basic characteristics in common. Let's take a look:

The amplitude of the wave (the height) determines the intensity of the wave (how bright the light, how loud the sound, or how likely the beach wave is to knock you down). The wavelength (λ - the Greek letter lambda) is the distance between two peaks of the wave or between two troughs (it’s the same). 

Although it can't be shown on this diagram, waves also have a frequency (ν - the Greek letter nu). Frequency determines how often a wave peak reaches a given point. If you are on the beach, the wave frequency is how many waves reach you per minute (or per hour...). Light waves are so small and travel so fast, that we measure the waves per second.

We can also discuss (but not show) the energy of the wave (E).

These three properties (λ, ν, and E) are related. We can begin to understand the relationship when we look at several different waves. The diagram below shows three different waves with different wavelengths.

As you can see the red light has the longest wavelength of the three and the blue has the shortest. Since all light travels at the same speed, then the visible part of each wave will pass by you in the same time. However, that means that 8 blue waves will pass you, but only 4 red waves. That means that blue light (with the smallest wavelength) has the highest frequency. This can be represented mathematically by the equation

Energy is directly related to frequency. Imagine that you were holding one end of a piece of rope (with the ohter end tied down). You could make wave patterns by swinging your end of the rope back and forth. If you wanted to make the blue pattern, you would have to wave your rope faster and more often than if you wanted to make the red pattern. That means that the blue pattern (with the higher frequency and smaller wavelength) requires more energy on your part. That relationship can be represented mathematically by the equation