Show that if the uncertainty in the location of a particle is about equal to its

Question

Show that if the uncertainty in the location of a particle is about equal to its de Broglie wavelength, that the uncertainty in its velocity is about equal to one tenth its velocity.

Explanation / Answer

The uncertainty principle (U.P.) says xp˜h-barwith h-bar=h/(2); h=Planck constant. Let x=, the De Broglie Wavelength as stated. Then, theU.P. says xp=p˜h/(2) Then, p=mv is the momentum and p/v=m--->p=mv so, U.P.---->xp=p=mv. From the DeBroglie relation, =h/p, so mv--->(h/p)mv, and again, p=mv so we have (h/p)mv--->h/(mv)mv˜h/(2), where the massm and Planck constant h cancels on both sides, so v/v˜1/(2)------->v=v/(2) Note that 1/(2)˜1/7˜1/10, thus v˜v/10, i.e. one tenth its velocity. Q.E.D. ---Hope this helps--- CHEERS!

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