midterm_review

# midterm_review - Here is a review of some(but not all of...

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Here is a review of some (but not all) of the topics you should know for the midterm. These are things I think are important to know. I haven’t seen the test, so there are probably some things on it that I don’t cover here. Hopefully this covers most of them. Vector Spaces Review properties on Shankar page 2 Closure under multiplication: If | u i and | v i ∈ V , then a | u i + b | v i ∈ V for any a, b . Note when b = 0 this takes care of scalar multiplication also. Inverses, identity, etc. Linear independence A set of vectors {| v i i} is linearly independent if a | v 1 i + b | v 2 i + · · · = 0 has only one solution: a = b = · · · = 0. Gram-Schmidt procedure If you have a set of linearly independent vectors | I i , | II i , . . . you can always construct an orthonor- mal set of vectors as follows: | 1 i = | I i p h I | I i | 2 i = | II i - | 1 ih 1 | II i normalization constant | 3 i = | III i - | 1 ih 1 | III i - | 2 ih 2 | III i normalization constant . . . The normalization constants are chosen so that h 2 | 2 i = 1 , h 3 | 3 i = 1 , . . . . Basis A basis of a vector space V is a set of vectors {| v i i} . 1

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Any vector | u i ∈ V can be written in terms of these vectors: | u i = a | v 1 i = b | v 2 i + . . . always has a, b, . . . so that the equation is satisiﬁed. Orthonormal (ON) basis An ON basis is one for which h v i | v j i = δ ij . Decomposition of unity If {| v i i} is an ON basis, then i | v i ih v i | = I . Linear Operators Linear operators have Ω e ( a | u i + b | v i ) = a Ω e | u i + b Ω e | v i . Operator Inverses The inverse of the product of operators is given by the inverses of those operators in reverse order: (Ω e Λ e ) - 1 = Λ e - 1 Ω e - 1 . Commutators
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midterm_review - Here is a review of some(but not all of...

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