Tag: WaveFunctions

In the last post, I showed you how a projectile in a superposition of moving to the left or of moving to the right can only be measured to be doing one or the other. But what happens to the wave function of the system when the measurement is made? Does it… does it… COLLAPSE!?

Sounds scary. But it is only scary when it is badly explained.

Today I’ll show you what wave function collapse would mean, what it would require, and what a couple of the alternatives are. Among other things, I’ll show you that:

• The standard way of explaining wave function collapse, which argues collapse is required to avoid a logical problem, is not legitimate;
• If the Schrödinger wave equation is correct, then wave function collapse can never happen (and anything resembling “collapse” is viewed not as a physical effect but as a user’s choice);
• Therefore, if wave function collapse really does occur, then the Schrödinger equation is wrong;
• But if the Schrödinger wave equation is correct, an understanding of why quantum theory predicts only probabilities for multiple possibilities, rather than definite outcomes, is still lacking.

Today’s post uses several previous posts and their figures as a foundation, so I’ll start with a review of the most recent one, with links to others of relevance.

What is a wave function in quantum physics?

Such a question generates long and loud debates among philosophers of physics (and more limited debate among most physicists, who tend to prefer to make predictions using wave functions rather than wondering what they are.) I have a foot in both camps, even though I have no real credentials among the former set. But no matter; today I won’t try to answer my own question in any profound way. We can debate the deeper meaning of wave functions another time.

Instead I just want to address the question practically: what is this function for, in what sense does it wave, and how does it sit in the wider context of physics?

Before we knew about quantum physics, humans thought that if we had a system of two small objects, we could always know where they were located — the first at some position x₁, the second at some position x₂. And after Isaac Newton’s breakthroughs in the late 17th century, we believed that by combining this information with knowledge of the objects’ motions and the forces acting upon them, we could calculate where they would be in the future.

But in our quantum world, this turns out not to be the case. Instead, in Erwin Schrödinger’s 1925 view of quantum physics, our system of two objects has a wave function which, for every possible x₁ and x₂ that the objects could have, gives us a complex number Ψ(x₁, x₂). The absolute-value-squared of that number, |Ψ(x₁, x₂)|², is proportional to the probability for finding the first object at position x₁ and the second at position x₂ — if we actually choose to measure their positions right away. If instead we wait, the wave function will change over time, following Schrödinger’s wave equation. The updated wave function’s square will again tell us the probabilities, at that later time, for finding the objects at those particular positions.

The set of all possible object locations x₁ and x₂ is what I am calling the “space of possibilities” (also known as the “configuration space”), and the wave function Ψ(x₁, x₂) is a function on that space of possibilities. In fact, the wave function for any system is a function on the space of that system’s possibilities: for any possible arrangement X of the system, the wave function will give us a complex number Ψ(X).

Drawing a wave function can be tricky. I’ve done it in different ways in different contexts. Interpreting a drawing of a wave function can also be tricky. But it’s helpful to learn how to do it. So in today’s post, I’ll give you three different approaches to depicting the wave function for one of the simplest physical systems: a single object moving along a line. In coming weeks, I’ll give you more examples that you can try to interpret. Once you can read a wave function correctly, then you know your understanding of quantum physics has a good foundation.

For now, everything I’ll do today is in the language of 1920s quantum physics, Schrödinger style. But soon we’ll put this same strategy to work on quantum field theory, the modern language of particle physics — and then many things will change. Familiarity with the more commonly discussed 1920s methods will help you appreciate the differences.