One Crazy Trick to Understanding Quantum Mechanics

As every physicist knows, the ability to understand quantum mechanics often depends on the ability to keep in mind that real physical quantities can be represented in equations by complex numbers, and thus by imaginary numbers.

Moreover, it is very useful to keep in mind, when plowing through quantum theory, to hold in your head the idea that the quantities in equations, whether real or complex, can be represented by infinities.

On the surface this sounds ridiculous, but it is quite true and quite practical. It means that everything that can be represented as a variable in an equation can be represented as an infinite series of terms added up.

You might think of it this way in concrete terms. If you are asked to measure someone's height, you might glance at them and say "about six feet". Six feet would be the first term in the sequence.

That might be a good enough answer. Then someone might ask you to be more precise, and you get out a tape measure and quickly ascertain "five foot ten."

So now you have two inches less than six feet. You have adjusted. The minus two inches is the second term of the series.

Then you are asked to measure again, more precisely, and you say "five foot ten and a quarter," so the plus quarter inch is the third term in the series.

And so on, getting more and more precise. In principle you could imagine an infinite number of terms, each smaller and smaller, added together to get an ever more precise measurement.

It's a great mental trick. to be able to look at an equation and see each variable as a sum of an infinite number of terms that way. As leas for me, the ability to see such an infinite series underlying all quantities can give me great ease in grasping an intuition of the computational power of the equations. 

It means that no matter how complicated the equation looks, I can remind myself of how to break it down into these little pieces. The key is that in physics, often you only care about the first couple terms in the series. You can't even measure any more precise than that.

For complex systems, physicists resort to infinite series so often to crank out real-world predictions that in physics the general approach is often called by the simple name analysis. 

In graduate school, we had to take a course in advanced analysis, called Mathematical Methods. My professor was a barrel-chested wild bearded hippie-looking type, a throwback to an earlier caveman type of physicist. When he got mad because his normal methods of solving an equation failed, he called the most brute force form of analysis getting out the hammer and tongs.