A spin-bond-order-wave (SBOW) state is characterized by unequal canonical ensemble averages of the bond-centered up- and down-spin densities, [<[c.sub.n:A(i)+][c.sub.n':A'(i')]>.sub.HF] [not equal to] [<[c.sup.[dagger].sub.n:A(i)-][c.sub.n':A'(i')-]>.sub.HF], where [<...>.sub.HF] means taking the quantum average in a HF eigenstate
. They are mathematically interesting but of no occurrence under any realistic parameterization .
In  we have proposed a new interpretation of the cosmic microwave background as a stable eigenstate
in a chain system of oscillating protons.
As an immediate application consider the number eigenstate
[[??].sub.n] = |n><n|, then
, the threshold level of disorder below which the probability of formation of an eigenstate
at the center of a PBG is negligible was established.
Berry  was the first who addressed this issue in quantum mechanics: He considered a system initially in an eigenstate
|n(R(t))>[.sub.t=0] of the governing Hamiltonian H(R(t)) dependent on the parameters R(t) changing with time t.
To this end, von Neumann proposed that the linear dynamics does not always apply; exactly when a measurement is performed on a system, the state of that system evolves nonlinearly and instantaneously to an eigenstate
of the observable being measured (von Neumann , V.1).
This is because the vacuum |0> is not the eigen state of [T.sub.00], but the eigenstate
of Hamiltonian H = [integral] [T.sub.00][d.sup.3]x (for the detailed discussions, see ).
In case where [absolute value of ([alpha])] = [absolute value of ([beta])]= 1/[square root]2 we see that the pure input state is an eigenstate
It follows that any macroscopic apparatus designed to measure any observable of any microscopic object will almost certainly get into an entangled quantum state with that object that is not an eigenstate
of the pointer observable of the apparatus.
In , for t < 0, the particle is in an eigenstate
of a semi-infinite square-well potential [V.sub.1](x),
Wave function collapse (WFC) is the phenomenon whereby a wave function, representing a quantum system and thus expressed in the linear sum of its eigenstate
basis, appears to single out one eigenstate
thus reducing the otherwise linear supposition into one single term.
This definition follows from [20, Corollary 1] which establishes that any single-qubit pure state not one of the six Pauli eigenstates
, together with Clifford group operations and Pauli eigenstate
preparation and measurement, allows universal quantum computation.