Many Worlds and the Measurement of an Electron

[PhyCurious]

TL;DR

How does many worlds deal with the measurement of an electron's position in space?

Summary: How does many worlds deal with the measurement of an electron's position in space?

Hi all - I am reading Sean Carroll's book on quantum mechanics and reached the end of the section on "branching and splitting" without getting an answer. I will lay out my assumptions and then get to the question.

An electron is a fundamental particle that is quantum in nature and described by a wave function. The wave function acts as something akin to a probability distribution for the observable being measured: e.g., if we're measuring spin-up or spin-down, prior to measurement the wave function has two branches, one where the electron is measured spin-up, the other spin-down; the amplitudes of these branches are √1/2, and the probability of observing each branch is the amplitude squared; hence, 1/2 for each branch.

Ok hopefully all of that is correct. Under the many worlds interpretation, when we conduct a measurement, the measuring device becomes entangled with the electron; via decoherence, the measuring device gets entangled with its environment, thus branching / splitting the wave function into two parts - one where the electron is in a spin-up state, the other where it is spin-down

Again, I hope this is all clear / correct. Now here's the crux of my question. What happens when we measure something that does not have two discrete parts, but rather is continuous? Let's take another example. We trap an electron in a 1m x 1m box. The wave function of the electron expands to fill the box; that is, when we eventually measure the position of the electron in space, it could be anywhere within the box. But just like there are infinite real numbers between 0 and 1, aren't there infinite positions within the box where the electron could be (or at least, 1m * the Planck length)? So we measure the position of the electron in the box, and get a definite point in space; doesn't the wave function split into an infinite number of branches based on the probability distribution, including some very tiny slivers of the wave function where the electron has tunneled through the box and escaped?

I don't think this thought experiment fundamentally challenges any aspects of MWI, but I'm curious as to whether I've understood it correctly and the opinions of the experts here. Thanks for the help!

 

[PeroK]

I think you've understood the idea. Any particular measurement with a particular piece of apparatus can actually only return a finite number of values. From that point of view, the split will be into a large but finite number of branches.

In theory, however, you may have interactions that result in an infinite number of possible results.

As I understand it, Everett who came up with MWI thought there would indeed an infinite number of branches.

PS I like your analysis of this, by the way.

 

[PhyCurious]

In theory, however, you may have interactions that result in an infinite number of possible results.

Interesting. Would this be an argument for an infinite, rather than a really large but finite, Hilbert space for our universe? Or does that not follow?

The whole concept of 'Hilbert spaces' is still really new to me, so trying to wrap my head around it...

 

[PeroK]

Interesting. Would this be an argument for an infinite, rather than a really large but finite, Hilbert space for our universe? Or does that not follow?

The whole concept of 'Hilbert spaces' is still really new to me, so trying to wrap my head around it...

Hilbert space itself is infinite dimensional. That's not really the issue. The mathematics of Hilbert spaces can handle whatever MWI wants to throw at it.

Here's an analogy that might help.

First, let's look at classical mechanics. You have a single particle. One way to study it is to create a "state space". This is a six-dimensional space which specifies the three position coordinates and the three velocity coordinates. If you think about it, all the information you need to describe the motion of a particle is contained in those six numbers. Its "state" tells you exactly where it is and in exactly which direction it is moving, at each point in time.

In QM the state space becomes the Hilbert space of wavefunctions. Now, a simple particle requires a continuous wavefunction to describe it. This is no longer just six numbers. It's a complex valued function of four variables (three spatial variables and time). Hilbert space is the mathematical space of all such functions.

One way to think about the infinite dimensionality of Hilbert space is to consider polynomials. Most functions can be approximated by polynomials. But, there are an infinite number of base polynomials:

$x, x^2, x^3 \dots x^n \dots$

And the list goes on indefinitely.

If you want to describe a wave function (or any function) you can specify an infinite polynomial (called a Taylor series). A typical wavefunction, at a single point in time in one dimension to keep things simple, would look like:

$\psi(x) = a_0 + a_1x + a_2x^2 + \dots$

That would be its Taylor series.

You can actually think of those coefficients $(a_0, a_1, a_2 \dots)$ as being the components of an infinite dimensional vector. Functions (including wavefunctions) are mathematically infinite dimensional vectors.

That's what it means to say that Hilbert space is infinite dimensional: you need an infinite number of coefficients to describe a typical wavefunction.

You can see from this that it is a very different beast from ordinary 3D space, where you need only three coordinates to specify a position: $(x_0, y_0, z_0)$.

 

[PhyCurious]

Very helpful, thank you!