Vector spaces &amp; representations

# Vector spaces &amp; representations - Correspondence...

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Correspondence between ordinary vectors and QM states: Vector Representation (in a particular basis) QM state Representation (in a particular basis) Vector A r Linearity: D A c C B A r r = r r r = + Column vector 3 2 1 a a a "ket" | Ψ > Linearity: χ = ϕ ψ = ζ + ϕ c Column vector ... 3 2 1 c c c Inner product operator: B r Row vector [] 3 2 1 b b b "bra" < ϕ | Row vector [ ] ... * * * 3 2 1 b b b Inner (dot) product: A B r r Result is a scalar. B A A B = r r r r Matrix product: 3 2 1 3 2 1 a a a b b b = b 1 a 1 + b 2 a 2 + b 3 a 3 = i i i a b "bracket" ψ ϕ = ψ ϕ dx * Result is a scalar. * ϕ ψ = ψ ϕ Matrix product: ... ... * * * 3 2 1 3 2 1 c c c b b b i i c b * =

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Basis vectors: z y x e e e ˆ , ˆ , ˆ ˆ , ˆ , ˆ 3 2 1 = = = = 1 0 0 ˆ 0 1 0 ˆ 0 0 1 ˆ z y x Basis states: | ϕ 1 >, | ϕ 2 >, … = ϕ = ϕ ... 0 1 0 ... 0 0 1 2 1 Orthogonality of basis vectors: j i e e z y z x y x j i = = = = 0 ˆ ˆ 0 ˆ ˆ ˆ ˆ ˆ ˆ [] 0 0 1 0 0 0 1 = , etc. Orthogonality of basis states: < ϕ i | ϕ j > = 0, i j 0 ... 0 1 0 ... 0 0 1 = , etc. Normalization of basis vectors: 1 ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ = = = = i i e e z z y y x x 1 0 0 1 0 0 1 = , etc.
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Vector spaces &amp; representations - Correspondence...

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