The uncertainty principle

where and are general functions.

Let
, where is an Hermitian operator,
and a general wave-function. We have

(205) |

where is the variance of [see Eq. (142)]. Similarly, if , where is a second Hermitian operator, then

(207) |

Now, there is a standard result in mathematics, known as the
*Schwartz inequality*, which states that

(208) |

Hence, if then Eqs. (206)-(209) yield

However,

(211) |

(212) |

(213) |

where

(215) |

Equation (214) is the general form of the *uncertainty principle* in quantum mechanics. It states that if two dynamical
variables are represented by the two Hermitian operators and ,
and these operators *do not commute* (*i.e.*, ),
then it is *impossible* to simultaneously (exactly) measure the two variables.
Instead, the product of the variances in the measurements is always greater than some critical value, which
depends on the extent to which the two operators do not commute.

For instance, displacement and momentum are represented (in real-space) by the
operators and
, respectively.
Now, it is easily demonstrated that

(216) |

(217) |

(218) | |||

(219) |

where is the momentum-space equivalent of .

Energy and time are represented by the operators
and , respectively. These operators do not commute,
indicating that energy and time cannot be measured simultaneously.
In fact,

(220) |

(221) |

(222) |

For instance, suppose that a particle passes some fixed point on the -axis.
Since the particle is, in reality, an extended wave-packet, it takes a certain amount
of time for the particle to pass. Thus, there is an uncertainty,
, in the arrival time of the particle. Moreover, since
, the only wave-functions which have unique energies
are those with unique frequencies: *i.e.*, plane-waves. Since a
wave-packet of finite extent is made up of a combination of plane-waves
of different wave-numbers, and, hence, different frequencies, there will
be an uncertainty in the particle's energy which is
proportional to the range of frequencies of the plane-waves making up the
wave-packet. The more compact the wave-packet (and, hence, the
smaller ), the larger the range of frequencies of the constituent plane-waves (and, hence, the large ), and
*vice versa*. To be more exact, if is the wave-function
measured at the fixed point as a function of time, then we can write

(223) |

(224) |

(225) | |||

(226) |

where . As before, Gaussian wave-packets satisfy the minimum uncertainty principle . Conversely, non-Gaussian wave-packets are characterized by .