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Unformatted text preview: 4. MATHEMATICAL FORMALISM 1. Vector Spaces. Dirac Notation 2. States as Vectors 3. Operators 4. Successive Operations. Commutators 5. Operators as Matrices 6. Expectation Values 7. More Theorems 8. Rules and Interpretations Problems This chapter needs a little more work, to explain a couple of things a little more thoroughly, and to discuss the “rules and interpretations” (Sec. 8) a bit more. However, this is it for now. Also, these subjects are left for Chapter 6: the uncertainty principle, Fourier transforms, momentum space, and mutually commuting observables. c ⃝ 2011, Charles G. Wohl, work in progress. 1 4 · 1. VECTOR SPACES. DIRAC NOTATION Vectors —In ordinary threedimensional, xyz space, we can set up a rectangular coor dinate system and write any vector a in the space in terms of three unit vectors, ˆ x , ˆ y , and ˆ z , that lie along the orthogonal axes: a = a x ˆ x + a y ˆ y + a z ˆ z . The orthonormal vectors ˆ x , ˆ y , and ˆ z form a basis for the space, and a x , a y , and a z are the components of a in this basis. The length a of a is given by a 2 = a 2 x + a 2 y + a 2 z = a · a , where a · a is the scalar product of a with itself. If a = 1, a is normalized. The scalar product of two vectors a and b is a · b = a x b x + a y b y + a z b z = ab cos θ , where θ is the angle between a and b . If a and b are perpendicular (or if a = 0 or b = 0), then a · b = 0: a and b are orthogonal. It is common and useful to write vectors using their components as the elements of column vectors. In order to more easily generalize, we shall label the axes 1, 2, 3 instead of x , y , z . The orthonormal basis vectors are ˆ 1 = 1 , ˆ 2 = 1 , ˆ 3 = 1 , and a is a = a 1 ˆ 1 + a 2 ˆ 2 + a 3 ˆ 3 = a 1 a 2 a 3 . Multiplication of a by a scalar c and addition of two vectors a and b are defined by c a = ca 1 ca 2 ca 3 and a + b = a 1 + b 1 a 2 + b 2 a 3 + b 3 . Any linear sum of vectors in the space, such as c 1 a + c 2 b + ··· , is a vector in the same space, because its three components are defined by the above operations. With vectors as the skinny matrices, we can use matrix multiplication to calculate scalar products if we tilt the first vector on its side. Thus a · b = ( a 1 a 2 a 3 ) b 1 b 2 b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3 . 2 The scalar product of, say, ˆ 3 with an arbitrary vector a picks out the third component of a : ˆ 3 · a = ( 0 1) a 1 a 2 a 3 = a 3 . Dirac notation —Paul Dirac invented a notation that, at the simplest level, can be used to distinguish row and column vectors. In his notation, a · b is written ⟨ a  b ⟩ . This is the scalar product of a bra (row) vector, ⟨ a  , and a ket (column) vector,  b ⟩ ; the product is a bra(c)ket . A scalar product—a bracket—is just a number, a simpler object than the vectors that make it up. Dirac notation makes marking vectors with bold print or an arrowvectors that make it up....
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 Spring '09
 Lee
 Linear Algebra, mechanics, Hilbert space, scalar product, Hermitian

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