Nonnegative quantum states

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

This corresponds to an anharmonic sequence of levels labeled by m (the total number of states being once more equal to N-l-1). This quantum number, m, should not be confused with the quantum number associated with the projection of the angular momentum operator. The most interesting situations occurs with the particular choice >li=0, A = A2 (i in (2.64) and (2.65). In this case it is possible to put the spectrum (2.65) in a one-to-one correspondence with the bound-state spectrum of the one-dimensional Morse potential. This can be done by choosing in Eq. (2.65) only the nonnegative branch of the quantum... 

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