Measuring Position of a Quantum Mechanical Particle

[chrisphd]

How would one measure the position of a quantum mechanical particle?

For example, suppose I am aware that an electron lies in a 1-D box of length L. And maybe I wish to know the position of the particle in that box to a certain level of accuracy. I'm satisfied with knowing the position within a range of L/2 for example. So I decide to divide the box into two separate boxes somehow. (Note: The outer edges of the two boxes will have an infinite potential ensuring that there can be no quantum tunnelling.)

Ok, so now i separate my two boxes. One box is transported to Antarctica, and the other to Mexico. I think that quantum mechanics would suggest that a wavefunction is coexisting in both the boxes prior to any further measurements being made. Firstly, could someone please tell me if the above sentence is correct, because it seems very bizaar to me that a wavefunction of a particle can be discontinous, which would be required when the boxes are separated.

Now my final question is, what experiment might I be able to do in order to force the electron to collapse into one of the two boxes. Or another way of posing my question is, how can I determine which box the "particle" will lie in.

 

[javierR]

You can certainly have a particle wavepacket that is peaked in macroscopically separated places. The overall wavefunction is not discontinuous; in your example where tunneling is prohibited, the wavefunction vanishes smoothly between the two boxes. I like to then use the consistent histories interpretation of quantum mechanics to understand these separated packets and other entanglement issues: http://quantum.phys.cmu.edu/CHS/histories.html

What experiment might I be able to do in order to force the electron to collapse into one of the two boxes. Or another way of posing my question is, how can I determine which box the "particle" will lie in.

Here's a related experiment that has been done (you could find other examples, but this is just one I remembered off-hand): http://www.aip.org/pnu/1995/split/pnu234-2.htm
It involves the sort of manipulations you're talking about:
"They find that by varying the relative phases of the laser pulses they can control whether the electron is on one side of its orbit or the other, a half micron away. Although they still possesses quantum properties, the electrons in a Rydberg wave packet state also behave in a sort of quasi-classical way like particles traveling in large elliptical orbits."

In your thought experiment, you might be able to invoke similar modifications so that you are modifying the wavepackets to make one region more unlikely than the other, perhaps even with zero amplitude in one box. Otherwise, without further modifications to your setup you can't determine which box the particle would be in without testing one of them.