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PhyCurious

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- 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!