Eigenvalues and Eigenvectors

(42) |

and have the property

(43) |

where , , are numbers called

Consider the eigenkets and eigenvalues of a Hermitian operator . These are denoted

where is the eigenket associated with the eigenvalue . Three important results are readily deduced:

*(i) The eigenvalues are all real numbers, and the eigenkets corresponding
to different eigenvalues are orthogonal.*
Since
is Hermitian, the dual equation to Equation (44) (for the eigenvalue
) reads

(45) |

If we left-multiply Equation (44) by , right-multiply the above equation by , and take the difference, we obtain

(46) |

Suppose that the eigenvalues and are the same. It follows from the above that

(47) |

where we have used the fact that is not the null ket. This proves that the eigenvalues are real numbers. Suppose that the eigenvalues and are different. It follows that

(48) |

which demonstrates that eigenkets corresponding to different eigenvalues are orthogonal.

*(ii) The eigenvalues associated with eigenkets are the same as the eigenvalues
associated with eigenbras.* An eigenbra of
corresponding to an eigenvalue
is defined

(49) |

*(iii) The dual of any eigenket is an eigenbra belonging to the same eigenvalue,
and conversely.*