Uncertainty Verification

I’m attempting to learn string theory, and I was reminded of this interesting tidbit I wrote up awhile ago. The uncertainty principle states that you cannot know position and momentum perfectly for a particle. Usually this is shown using commutation relations and you end up with an actual inequality: \Delta x\Delta p\geq \frac{\hbar}{2}.

I wondered what happened mathematically if we knew the position of a particle. Say the wavefunction of the particle was given by \psi=\delta(x). This “delta function” says that the particle is precisely at the point in space x=0. I won’t go into the details of the function, since they are numerous. The important thing is that if we can calculate the momentum in any sense at all, then we will have violated the uncertainty principle.

Let’s look at the expected value of the momentum. E(p) = \int_{ - \infty}^\infty \delta(x)\left( - i\hbar\frac {d}{dx}\right)\delta(x) dx = - i\hbar\int_{ - \infty}^\infty \delta(x)\delta'(x)dx. But what is that integral? It is some constant multiple of \delta'(0) which doesn’t exist (I glossed over it, but it is technically because the delta function is not a function but a distribution). This confirms the uncertainty principle, because if we know the position, then the momentum is so badly behaved that it doesn’t even have an expected value (and hence not a definite value).

Edit: Also, I’m having major issues figuring out how to make angle brackets when typing math.


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