# Download Circuit Analysis (Horwood Engineering Science Series) by John E. Whitehouse PDF

By John E. Whitehouse

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Example text

32) where eiαx(τ ) = e−iτ H0 eiαx eiτ H0 . Hence we have proved the formula ⎡ ⎤ ⎢ Ω0 (x) ⎣ − 12 | cos At| Rn ∼ e i 2 t (γ˙ 2 −γA2 γ)dτ i e 0 n αj γ(tj ) j=1 ⎥ Ω0 (γ(0))dγ ⎦ dx γ(t)=x = Ω0 , eiα1 x(t1 ) . . 33) for 0 ≤ t1 ≤ . . 3 Let now H = H0 + V (x) . 34) We have the norm convergent expansion e−itH = ∞ (−i)n n=0 V (t1 ) . . V (tn )e−itH0 dt1 . . ≤tn ≤t where V (τ ) = e−iτ H0 V eiτ H0 . 16) can be taken in the strong sense in F (H0 ), that ⎡ ⎢ f (x)Ω0 (x) ⎣ 1 | cos At| 2 Rn ∼ e γ(t)=x i 2 t 0 (γ˙ 2 −γA2 γ)dτ −i e t V (γ(τ ))dτ 0 ⎤ ⎥ g(γ(0))Ω0 (γ(0))dγ ⎦ dx = (Ω0 , f e−itH gΩ0 ) .

55) that W−∗ (δ, x), as a function of δ for ﬁxed x, is the asymptotic probability amplitude in momentum space as t → +∞ of a particle located at x for t = 0. 56) dγ . 57) may also be deﬁned by series expansion of the second term in the integral. 59) Rn where we have taken = 1 and the integration is to be understood in the weak sense. 60) γ(t) t =V± where γ0 (τ ) = v− · τ ∧ 0 + v+ · τ ∨ 0 + y are the asymptotes of γ(τ ), S(γ) is the action along the path γ and S0 (γ0 ) is the free action along the asymptotic path γ0 .

Tn ≤t where V (τ ) = e−iτ H0 V eiτ H0 . 16) can be taken in the strong sense in F (H0 ), that ⎡ ⎢ f (x)Ω0 (x) ⎣ 1 | cos At| 2 Rn ∼ e γ(t)=x i 2 t 0 (γ˙ 2 −γA2 γ)dτ −i e t V (γ(τ ))dτ 0 ⎤ ⎥ g(γ(0))Ω0 (γ(0))dγ ⎦ dx = (Ω0 , f e−itH gΩ0 ) . 21) below. 38) for all t such that cos At is non singular and V ∈ F(Rn ) and the initial condition ψ(x, 0) = ϕ(x) is in F (Rn ) ∩ L2 (Rn ). 37) for all ϕ ∈ F(Rn )∩L2 (Rn ). e. by any path γ0 (t) for which t γ0 (t) = 0 and the kinetic energy 12 0 γ˙ 0 (τ )2 dτ is ﬁnite.