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CRC Press eBooks

The Adiabatic Approximation

Şaki̇r Erkoç

Published Jul 28, 2006• 0 citations
Publisher

Abstract

Consider a perfect pendulum oscillating in a vertical plane, which is supported in a box (see Figure 17.1 ). Fig. 17.1 An oscillating pendulum in a box. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429136528/397e01c1-e101-4a47-b552-a910642bbb8c/content/fig17_1_B.tif"/>

DOI
10.1201/9781584887331-17

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Q24(1) Consider a three-dimensional Hilbert space ℋ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1209.tif"/> with a preferred decomposition as the direct sum of a complete orthogonal family of one-dimensional subspaces S → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1210.tif"/> , S → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1211.tif"/> and S → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1212.tif"/> , i.e., ℋ → = ℋ → ⊕ = S → ( − ) ⊕ S → ( 0 ) ⊕ S → ( + ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0459.tif"/> Let η → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1213.tif"/> , η → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1214.tif"/> and η → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1215.tif"/> be unit vectors in S → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1216.tif"/> , S → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1217.tif"/> and S → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1218.tif"/> , respectively. 1 A vector in ℋ → ⊕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1219.tif"/> is of the form η → ⊕ = c − η → ( − ) ⊕ c 0 η → ( 0 ) ⊕ c + η → ( + ) = c − η → ( − ) ⊕ + c 0 η → ( 0 ) ⊕ + c + η → ( + ) ⊕ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0460.tif"/> where C − , c 0 , c + ∈ ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1220.tif"/> .

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Consider a perfect pendulum oscillating in a vertical plane, which is supported in a box (see Figure 17.1 ). Fig. 17.1 An oscillating pendulum in a box. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429136528/397e01c1-e101-4a47-b552-a910642bbb8c/content/fig17_1_B.tif"/>
Q24(1) Consider a three-dimensional Hilbert space ℋ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1209.tif"/> with a preferred decomposition as the direct sum of a complete orthogonal family of one-dimensional subspaces S → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1210.tif"/> , S → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1211.tif"/> and S → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1212.tif"/> , i.e., ℋ → = ℋ → ⊕ = S → ( − ) ⊕ S → ( 0 ) ⊕ S → ( + ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0459.tif"/> Let η → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1213.tif"/> , η → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1214.tif"/> and η → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1215.tif"/> be unit vectors in S → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1216.tif"/> , S → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1217.tif"/> and S → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1218.tif"/> , respectively. 1 A vector in ℋ → ⊕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1219.tif"/> is of the form η → ⊕ = c − η → ( − ) ⊕ c 0 η → ( 0 ) ⊕ c + η → ( + ) = c − η → ( − ) ⊕ + c 0 η → ( 0 ) ⊕ + c + η → ( + ) ⊕ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0460.tif"/> where C − , c 0 , c + ∈ ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1220.tif"/> .