Population operators

In other words the state , f)s is an eigenstate of the operator Nx with eigenvalues nk Nx is called the population operator for the A state. 

The state w, f>s is an eigenstate of N with eigenvalue N, and N is called the total population operator. Because the vector , Os is a function of the time, it is necessary to specify the time at which the creation or annihilation operators are applied, and in some discussions it may be advisable to indicate the time explicitly in the symbol for the operator. For our present discussion it will be sufficient to keep this time dependence in mind. In an expression such as Eq. (8-109), all the creation operators are applied at the same time, and since they all commute, this presents no logical problem. The order of the operators in the definition Eq. (8-107) is important however the opposite order produces a different operator ... 

The Heisenberg equation for the population operator is coupled to analogous equations for the free-bound coherences with n) = 1), 2) ... 

They define the corresponding electron population operators, and = Np,... 

Alternatively, W[Qa] can be calculated as the expectation value of the basin population operator introduced by Diner and Claverie [116] ... 

The eigenvalues, W[ 2a1, of IV [Da] belong to the series of integer 0,..., N and represent all the accessible numbers of electrons within Da- As a consequence of 23, the eigenvalues of the population operators of different basins are correlated since they also obey the closure relation... 

The closure relation of the basin population operators enables to carry out a statistical analysis of the basin populations through the definition of a covariance matrix [67]. The covariance operator is a matrix operator in which elements are deduced from the classical expression of the covariance ... 

This is a method of identifying errors and unsafe conditions that contribute to both potential and actual accidents or incidents within a given population by means of a stratified random sample of participant-observers selected from within the population. Operational personnel can collect information on potential or past errors or unsafe conditions. Hazard controls are then developed to minimise the potential error or unsafe condition. 

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