How do you visualize an eigenfunction or eigenvector? Today I present a very physical way to think about these ideas.
If you’re reading this, it’s not by accident. You’ve probably seen these in a class or a textbook already. Your prof probably told you that an eigenfunction was a transform matrix A that, when applied to an eigenfunction v, will produce the same function times an eigenvalue (a constant) λ. You might’ve done it a couple times in homework and had to flip to the glossary several times until you figured out what they did. But if you’re like what I was like in second year, which was a highly visual thinker who preferred seeing things ideas action, you might not have a physical context for what eigenfunctions and eigenvalues mean. In fact it wasn’t until the next year that I had my big “OHHHH” moment in a quantum mechanics class, but these are important in most science and engineering settings.
Big idea of this post: Eigenfunctions and eigenvalues describe measurement. You can think of the system you’re trying to measure as a function (kind of like v) and the measurement you’re making as a transform matrix B (like A). If at the end of your measurement your system is still the same as it was at the beginning, you’ll get a result φ (like λ) and your original system. If this is true, meaning if the system at the end of the measurement HAS NOT CHANGED since before the measurement started, then the matrix B is an eigenfunction and the result φ is an eigenvalue. If the system is different, you’ll still get a result φ, but if you were to apply the same measurement again you would get a different answer.
If this sounds like a familiar idea, than you might just be familiar with one of the most important ideas of quantum mechanics, in that sometimes measuring a system will change it. In QM, you see eigenfunctions and eigenvalues all the time.
Let’s put this in a more general setting.
In our experiment, we’re trying to measure the amount of adrenaline in someone’s body. We do this by taking a blood sample in a clinic.
The first person to come to our clinic is Vivian. Vivian’s mom is diabetic, so she’s fine with needles because she’s seen them all her life. Our nurse is able to take a blood sample and determine that there’s no adrenaline in her body. If she were to take a second sample, Vivan would still be calm and cool, so the nurse would get the same result: no adrenaline; calm Vivian.
The next person to come to the clinic is Olivier. Olivier hates needles (he’s doing the experiment to try and get over his fear). When the nurse takes his blood sample, she does it quickly before he has time to think about it, but the anticipation in the lobby made his adrenaline level rise a little bit. Now his brain has registered that he’s getting injections, so his body ramps up his production of adrenaline. If the nurse were to take a second blood sample, it would show even more adrenaline in the results, and Olivier would be much less calm than he was at the start.
Our measurement in both cases is the same, so we represent it with the symbol N (for needle). We also have two initial functions; Vivian (v) and Olivier (o). No matter how many times Vivian’s blood is taken, this measurement looks like Nv=φv. φ is always the same and v is always the same. This means that Vivian is an eigenfunction of the matrix N and her blood-adrenaline level is an eigenvalue φ. But when Olivier goes up, his measurement looks like No=εö. Not only does ε change, but now o has turned into ö (which represents freaked-out Olivier); the end function is a different state than o was originally. So o is not an eigenfunction of N, and ε is not an eigenvalue.
This demonstrates another important concept: The eigenvalues of an eigenfunction A will not always be the same eigenvalues if A is used on different vectors. N acting on v produced eigenvalues, but N acting on o did not. If we were to act N on a third person (let’s call them Urie, u) who is also calm around needles, he might have a different adrenaline level ξ than Vivian’s φ. So u is also an eigenfunction of N, but the eigenvalue ξ is unique to u.