We are unable to measure the exact position and momentum of a particle due to uncertainty principle. If we want to measure both the position and the momentum of a electron, we will shine light of wavelength less than the size of electron but in doing so we will change the momentum of the electron to a very large extent as the photons have high energy. But if we shine light from two opposite directions, can we measure both the position and momentum accurately, at least theoretically?
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$\begingroup$ Surely no, but I can not answer, why not. $\endgroup$– b.LorenzCommented Jan 11, 2017 at 18:47
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1 Answer
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With each individual measurement, from each individual photon that scatters off the electron, you can have your equipment set up to measure either the position, or the momentum, but not both.
If you cannot measure position and momentum at the same time with any single photon, you cannot measure it at the same time with any amount of photons, no matter how many photons you use, or how many directions you shine them from.
Furthermore, the wavefunction that describes the electron is disturbed with each measurement. Its shape will change (or at least appear to, depending on which interpretation of Quantum Mechanics you choose to believe).
The maths that describes that shape means that the smaller the variation in possible momentum that is encoded within the shape, the larger the variation in possible position.
Mathematically speaking, the shape of wavefunction that has the smallest possible variation in both momentum and position is the Gaussian wave packet, which you can read about at https://en.wikipedia.org/wiki/Wave_packet
It is this shape that gives the lower bound in uncertainty described by the famous uncertainty principle.
