Eigenstates and Eigenvalues

When we discussed a standing wave on a string we saw that it was possible to set up waves with any number of vibrating segments of equal length. Here’s the picture showing four possible ways to make a string vibrate. You can imagine what the other ways look like- just keep increasing the number of segments.

Each of the four wave patterns has its own rate of vibration, or frequency, which is proportional to the number of vibrating segments. A wave with three segments has a frequency that is three times higher than a wave with one segment.

When you make a string vibrate in one of these patterns it is called a pure frequency state, or an 'eigenstate' of frequency. The frequency of the eigenstate is called the frequency 'eigenvalue'. The prefix 'eigen' is a German term meaning intrinsic.

In real life if you pluck a guitar string what actually happens is that some mixture of all these pure waves, or eigenstates, start vibrating together, and the overall pattern of movement can be very complicated. If you ask what is the frequency of the resulting hybrid wave, the answer is that it no longer has a single frequency- it is vibrating with a mix of frequencies all mingled together.

A very similar effect occurs in quantum theory. In the same way that a guitar string can vibrate in one of many distinctive eigenstates, each of which has a well-defined frequency, so the wave of a particle can be one of several eigenstates of the particle, each with a well-defined energy. But just as a guitar string can also vibrate in a hybrid way as a jumble of eigenstates with no well-defined frequency (in fact that's how guitar strings usually vibrate), so the wave of a particle can be a jumble of energy eigenstates with no well-defined energy.

What happens when you measure the energy of a particle in a jumbled state is that the wave of the particle switches from being a jumble of different eigenstates into being a single eigenstate with a single energy. It is exactly as if  you had roughly plucked a guitar string to make it vibrate in a jumbled way, and when you tried to measure the frequency of the jumbled vibration the guitar string instantly started to vibrate in one of the pure frequency states rather than vibrating in a jumbled way!