Heisenberg Uncertainty Principle

(2.91) |

where use has been made of Equation (2.78) (for the case of a system with one degree of freedom). The solution of the previous differential equation is

(2.92) |

where . It is easily demonstrated that

(2.93) |

The well-known mathematical result [92]

yields

(2.95) |

This is consistent with Equation (2.80), provided that . Thus,

Consider a general state ket whose coordinate wavefunction is , and whose momentum wavefunction is . In other words,

(2.97) | ||

(2.98) |

It is easily demonstrated that

and

(2.100) |

where use has been made of Equations (2.27), (2.81), (2.94), and (2.96). Clearly, the momentum space wavefunction is the

Consider a state whose coordinate space wavefunction is a *wavepacket*.
In other words, the wavefunction only has non-negligible amplitude in some
spatially localized region of extent
. As is well known, the Fourier
transform of a wavepacket fills up a wavenumber band of approximate extent
. [51]. Note that, in Equation (2.99), the role of the wavenumber
is played by the quantity
. It follows that the momentum space
wavefunction corresponding to a wavepacket in coordinate space extends over
a range of momenta
. Clearly, a measurement
of
is almost certain to give a result lying in a
range of width
. Likewise, measurement of
is almost certain to
yield a result lying in a range of width
. The product of these two
uncertainties is

(2.101) |

This result is called the

Actually, it is possible to write the Heisenberg uncertainty principle more exactly by making use of Equation (1.86) and the commutation relation (2.47). We obtain

(2.102) |

for a general state. It is easily demonstrated that the minimum uncertainty states, for which the equality sign holds in the previous relation, correspond to Gaussian wavepackets in both coordinate and momentum space. (See Exercise 4.)