Heisenberg's Uncertainty Principle

(1134) |

Furthermore, we can interpret and as characterizing our uncertainty regarding the values of the particle's position and wavenumber, respectively.

A measurement of a particle's wavenumber, , is equivalent to a measurement of its momentum, , because . Hence, an uncertainty in of order translates to an uncertainty in of order . It follows, from the previous inequality, that

This is the famous

It is apparent, from Equation (1133), that a particle wave packet of initial spatial extent spreads out in such a manner that its spatial extent becomes

at large . It is readily demonstrated that this spreading of the wave packet is a consequence of the uncertainty principle. Indeed, because the initial uncertainty in the particle's position is , it follows that the uncertainty in its momentum is of order . This translates to an uncertainty in velocity of . Thus, if we imagine that part of the wave packet propagates at , and another part at , where is the mean propagation velocity, then it follows that the wave packet will spread out as time progresses. Indeed, at large , we expect the width of the wave packet to be

(1137) |

which is identical to Equation (1136). Evidently, the spreading of a particle wave packet, as time progresses, should be interpreted as representing an increase in our uncertainty regarding the particle's position, rather than an increase in the spatial extent of the particle itself.