ssp_0a - 0. Elementary quantum mechanics 0.1 Mathematical...

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1 0. Elementary quantum mechanics 0.1 Mathematical tools Vector space of the quadratically integrable complex-valued functions (29 fx o r -a 2 dx b <∞ ∫∫ ... infinite-dimensional! n-dim. vector space V n Hilbert space H aa a a in = 1 ,..., ,. .., Index = 1,. .., Sum Argument x axb << mostly -∞ < < ∞ x ab c += (29 (29 (29 fx gx hx cab ii i =+ Addition of functional values for each argument x Scalar product a b i n ⋅= = 1 for V n real (29 (29 fg f x gx dx = -∞ * often = ... i n = * 1 for V n complex
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2 Scalar product Length (norm) (29 aa a a i i n =⋅ = = 2 1 1 2 ff f f x d x == -∞ 2 1 2 Orthogonality ab a b ⊥⇔ = 0 Parallelity a b || ⇔= α with scalar α fg f g = |0 f g || Properties of the scalar product ( 29 ( ( αα gf g f g f g |* | * ||| = = += + 12 1 2 f g f g f g | +≤ + -≥ - (Schwarz inequality) (triangle inequality)
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3 Linear matrices (29 M M M MM M = == = += + = mm ab b m a aM a a b n nn n ii j j n j 11 1 1 1 K K , αα e.g.,
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ssp_0a - 0. Elementary quantum mechanics 0.1 Mathematical...

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