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this post was submitted on 11 Aug 2024
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There are an infinite number of velocities I can use to get up off the couch right now.
That does not mean that I will get up off the couch at infinite velocity.
Yes, there are an infinite number of velocities you can use, but if you look at their distribution, you'll find that it quickly goes to zero somewhere around 1-2 m/s, so the expectation value of the velocity is convergent.
If you have an object with a velocity taken from a distribution that doesn't approach zero sufficiently fast as the velocity goes to infinity, the expectation value diverges. A simple example would be a person that would be half as likely to get up at a velocity of 2 m/s as 1 m/s, and half as likely to get up at 4 m/s as 2 m/s, etc.
The more mathematical version of the same argument is to compute the kinetic energy of a particle whose wavefunction is a delta pulse (i.e. a particle whose position is exactly defined), and you'll find that the particle has infinite energy.
You choose a velocity from an infinite number of options, but the electron exists in a superposition of all those options.