(68) |

where the expectation value is taken over the particular physical state under consideration. It is obvious that the expectation value of is zero. The expectation value of is termed the

(69) |

The variance of is a measure of the uncertainty in the value of for the particular state in question (i.e., it is a measure of the width of the distribution of likely values of about the expectation value). If the variance is zero then there is no uncertainty, and a measurement of is bound to give the expectation value, .

Consider the Schwarz inequality

(70) |

which is analogous to

(71) |

in Euclidian space. This inequality can be proved by noting that

(72) |

where is any complex number. If takes the special value then the above inequality reduces to

(73) |

which is the same as the Schwarz inequality.

Let us substitute

(74) | ||

(75) |

into the Schwarz inequality, where the blank ket stands for any general ket. We find

where use has been made of the fact that and are Hermitian operators. Note that

(77) |

where the

(78) | ||

(79) |

The commutator is clearly

(80) |

whereas the anti-commutator is obviously Hermitian. Now, it is easily demonstrated that the expectation value of an Hermitian operator is a real number, whereas the expectation value of an anti-Hermitian operator is an imaginary number. It follows that the right-hand side of

(81) |

consists of the sum of an imaginary and a real number. Taking the modulus squared of both sides gives

(82) |

where use has been made of , etc. The final term on the right-hand side of the above expression is positive definite, so we can write

where use has been made of Equation (76). The above expression is termed the