(Re)normalization: an Introduction

A statistical interpretation of the wavefunction

If we wanted to answer the question of what’s truly fundamental in this Universe, we’d need to investigate matter and energy on the smallest possible scales. On such scales, reality starts behaving in strange, counterintuitive ways. We can no longer describe reality as being made of individual particles with well-defined properties like position and momentum. Instead, we enter the realm of the quantum: where fundamental indeterminism rules, and we need an entirely new description of how nature works.

Ψ. Meet every quantum physicist’s best friend — the 23rd Greek letter. Ψ (psi, pronounced with the /ps/ in “lapse” followed by an /eye/) represents the wave function of a quantum system. We derive our formalism of Ψ from Schrodinger’s equation:

Schrödinger’s equation can be written in many forms. In its simplest one, it looks like this.

This daunting-looking equation is where we stand. It is by any measure, the most successful quantum mechanical postulate of all time. We won’t be delving into this for today. If you’re curious, check this out:

Schrödinger’s Equation: Explained

Now, wavefunctions. What are they? I have dedicated a 7-minute read if you’d like to know more about it but briefly explained, a wavefunction is a mathematical description of everything we know about a particular quantum system. A quantum system is of course a “system” we happen to be studying using quantum mechanics. It could be anything.

All you need to know about the Schrodinger equation for this article is that it plays a role logically analogous to Newton’s second law. That is to say, given appropriate initial conditions, the Schrödinger equation determines Ψ(x, t) for all time — just as in classical mechanics how Newton’s law determines x(t) for all future time.

But what exactly is a “wave function”, and what does it do once you’ve survived the math needed to derive it? After all, a particle, as the name suggests, is localized to a point in space. The very nature of matter is such that you don’t have probabilistic realities. You don’t have a bed that might not be there when you go to sleep at night — at least I hope not. Those odds would be awful.

Wavefunctions, as will be apparent, describe probabilistic positions of a quantum state over a region of space. The answer to “why wavefunctions?” is given by Born’s statistical interpretation.

Modulus of psi squared

Born’s interpretation says that the integral of |Ψ(x, t)|² dx is equal to the probability of finding a particle at point x at time t. Naturally, the statistical interpretation introduces an element of indeterminacy to quantum mechanics. From a philosophical perspective, this is a problem. From a physical one? Not so much. All quantum mechanics has to offer is statistical results. Certainty is thrown out the window. Seriously, scientists are just a bunch of idiots. It’s all God.

In all seriousness, if the integral |Ψ|² is equal to the probability of finding a particle at point x, then it must follow that this quantity is equal to one (the particle’s got to be somewhere). This is where normalization introduces itself.

Without this, Born’s statistical interpretation would be near nonsense. Normalization is the scaling of wave functions so that all the probabilities add to 1. The probabilistic description of quantum mechanics makes the best sense only when this holds. This requirement should bother you. The wave function must be determined by the Schrodinger equation and to impose an extraneous condition on Ψ without checking its consistency would be incredibly stupid of us. So we invoke the following idea.

If the integral of Ψ is not 1 and is instead equal to some other constant, we incorporate that constant into the wave function to normalize it to scale the probability to 1 again. For some solutions of the Schrodinger equation, it happens such that the integral is infinite; in this case, no multiplicative factor will normalize it back to one. Such solutions are said to be trivial in the sense that they cannot represent a particle and must be rejected. These are non-normalizable solutions.

Max Born: center. Credit.

But there comes the natural concern. Suppose I do have a normalized wave function at t = 0. How do I know that the wave function will remain normalized for all time as Ψ evolves? You can’t keep normalizing the wave function — you would need a function that normalizes Ψ. This would not be a solution to the Schrödinger equation anymore. We would quickly descend into nonsense.

Fortunately, the Schrödinger equation mathematically preserves the normalization of a wave function. Without this, we’d have something incompatible with the Born interpretation and all of quantum mechanics would crumble. Here’s hand-wavy proof most of you might find in introductory quantum mechanics textbooks. For those of you who care, check it out. Otherwise, I’ll end it here.

As always, thank you for reading!