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100) It is possible to calculate the expectation values of a; , using the formula = Tr(pxa;). >= 0, = coswt, and = sinwt, as are expected from the same result as in Eq. 101) . Both Px(t) and P-x(t) are pure-state density matrices, and satisfy the condition = 1 and p2 = p. 102) ° This is not a pure-state density matrix. It satisfies the trace relation Tr(p) = 1, but Tr(p 2 ) is less than one. 3 illustrates. 3 on the rotating wave approximation. If the spin is along the z direction initially, Pn(t) = 1- P22(t) = pu(t) = p; 1 (t) = -i (~) 2 (sin(nt/2)) 2 , (~) ei(w-a)t (sin(Qt/2)) x { cos(fH/2)- i (w; a) sin(nt/2)}.

If the system is in the state 1/J( x) and an observation is made as a result of which the system's state vector becomes ¢>( x ), the probability of this result of the observation is I ( 1jJ, ¢>) 12 , which is the absolute square of the scalar product of the two state vectors. In terms of the Wigner functions, the transition probability can be written as I (1/J,r/>) 12 = 27r j W,p(x,p)Wq,(x,p)dxdp. 12) This expression is a clear proof that the Wigner function cannot be positive everywhere in phase space.

There are many other approaches to this problem. The rotating wave approximation is one of them. 27) 24 Forms of Quantum Mechanics then Eq. 28) These equations appear very simple, but cannot be solved analytically. It is a simple matter these days to solve these equations numerically. This does not prevent us from studying the properties of these equations. 29) Next, it should be noted that the difference between the two frequencies (w- a) can be much smaller than the sum (w +a). Thus, if we are primarily interested in the frequency region where (w- a) is small, the high frequency component of (w + a) can be ignored.

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Classical and quantum mechanics of the damped harmonic oscillator by Dekker H.

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