Heisenberg Uncertainty Principle scientific prove [migrated]

Heisenberg's uncertainty principle states that: if the x-component of the momentum of a particle is measured with an uncertainty $$\Delta \vec p_x$$ then its x-position cannot, at same time, be measured more accurately than $$\Delta\vec x=\frac {\hbar}{2\Delta\vec p_x }$$ $$\Delta\vec x\Delta\vec p_x \ge \frac {\hbar}{2}$$ what is scientific prove of this principle?

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 This is a standard textbook question, so I am moving it to physics.SE. Moreover, when refer to a component of a vector then you drop the arrow. – Piotr Migdal♦ Apr 20 '12 at 16:15