Thought experiment about superposition.

[rasp]

Suppose I could restrict an electron to a very small space byfor example using multiple magnetic fields, then could I not be sure with 100% probability that the electron was there before and after a measurement? Wouldn’t such a experimental set up dis prove the idea that the electron had freedom of motion before the measurement?

 

[BvU]

I suppose not. In QM there is the case of the delta function potential well

 

[king vitamin]

Suppose I could restrict an electron to a very small space byfor example using multiple magnetic fields, then could I not be sure with 100% probability that the electron was there before and after a measurement?

You cannot do so to a 100% certain size. If you continue to localize the electron, the energy you would require to do so would spontaneously create more particles out of the vacuum and cause more uncertainty in your experiment. .

In fact, I actually don't need to cite the relativistic answer (relevant to our universe) to invalidate your conclusions. If you approximate your magnetic fields as an infinite potential well in non-relativistic quantum mechanics, then by localizing your magnetic fields you must input increasing energy to support an electron in its ground state. As you can see from the explicit value for the energy of an electron in the ground state of an infinite well, this energy increases as 1/L^2 with decreasing size L, so the energy you need to input to make position 100% localized diverges to infinity.

 

[Nugatory]

Suppose I could restrict an electron to a very small space by for example using multiple magnetic fields, then could I not be sure with 100% probability that the electron was there before and after a measurement? Wouldn’t such a experimental set up disprove the idea that the electron had freedom of motion before the measurement?

There is a somewhat related discussion starting at post #3 of https://www.physicsforums.com/threads/evolution-of-the-wavefunction.960445/ and consider especially the negative delta function case: $V(x)=-\delta(x-X_0)$.

You also want to be very cautious about phrases like "the electron had freedom of movement before the measurement"; these will tempt you into classical assumptions about how electrons behave, and that's just not how it is. Quantum mechanics actually says something more along the lines of "this is the probability of finding it this far from where we last found it", and there is no physically realizable way of reducing the "this far" to zero.

Finally, the physically unrealizable state in which the next position measurement is guaranteed to yield the same result as the last one is a superposition of momentum states... so even if you could realize that single definite position state, you still wouldn't have disproved superposition.