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It seems quite reasonable to consider also approximate values for the observables of a quantum system. For example, we can consider a Gaussian distribution of the position and wavelength, λ = h/p, of a wave packet with mean: x¯ and variance σ, and, respectively, p¯. Then the wave function: φ(x) = (πσ 2 )−1/4 exp x − x¯ 1 + i¯ px 2 2σ is a minimum uncertainty wave packet with: 1 ∆x = √ σ, ∆p = √ σ and ∆x ∆p = σ 2 . 2 2 2 Minimum uncertainty wave functions can be extended to non-commuting operators other than position and momentum [325].

The probability that the system is in state | 0 is one at the initial moment, t = 0, decreases to zero and then oscillates between one and zero in time. We say that the magnitude of the two probability amplitudes varies harmonically 3 with time. The probability to find the system in one of the two states varies back and forth between the magnitudes of the two individual probabilities. 3 A harmonic variation is expressed in complex exponential form as Aei(ωt+ ) , where ω = 2πν is the angular velocity.

Different mixtures of pure states could have the same density matrix; for example, the mix of the pair of states 1 | + = √ (| 0 + | 1 ) 2 and 1 | − = √ (| 0 − | 1 ) 2 with probability p = 1/2 and the mix of pair of states | 0 and | 1 with probability p = 1/2 have the same density matrix. 3 where we show that the density matrix allows us to distinguish pure states from mixed (impure) states: tr (ρ2 ) = 1 for a pure state and tr (ρ2 ) < 1 for a mixed state. Pure states are represented as points on the surface of the Bloch sphere while mixed states are points inside the Bloch ball.

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