By Professor Crispin W. Gardiner Ph. D (auth.)
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The terms corresponding to frequencies significantly different from ±n will oscillate so rapidly that their net effect will be almost zero. We can even evaluate the correlations of E(t), E(t'), under this approximation. 5,6), but 16 1. A Historical Introduction omit the factor "1, which refers specifically to the interaction. Thus we assume that the electric field operator behaves like our "quantum noise". 19). These correlation functions are of exactly the same form as used in the arguments for quantum mechanical consistency of Sect.
We will consider a system describable as above, by the eigenstates la}, and assume that these have a discrete spectrum. It is obviously possible to approximate any situation by this kind of description, since a discrete set can always be ordered, and hence described by a single operator whose eigenvalues represent the position in order. 26 2. 1), that the final state is given up to a normalisation factor by a linear operator acting on it. With an imprecise measurement there is some arbitrariness in defining what the measured value obtained by a measurement actually is.
7) a p is positive semidefinite: for any state IA} (AlpIA) = LPal(AI1PaW 2:: O. 9) and this is said to be a pure state, since the system is always in the state l1Pa}. 9). 11) where Ai are the eigenvalues of p, and Ii} is the corresponding eigenvector. Because in this case p2 = p, only Ai = 0, I are allowable, and because Tr {p} = 1, only one eigenvalue, say Am is 1. 9). 13) d) with equality only for a pure state. 16). 21 ) for some an . 22) that is, a pure state. 24 2. 23) as . 24) where H is the Hamiltonian for the system.