By Nadir Jeevanjee
An advent to Tensors and workforce idea for Physicists presents either an intuitive and rigorous method of tensors and teams and their function in theoretical physics and utilized arithmetic. a selected target is to demystify tensors and supply a unified framework for knowing them within the context of classical and quantum physics. Connecting the part formalism familiar in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings could be a great addition to the literature on tensors and crew theory. Advanced undergraduate and graduate scholars in physics and utilized arithmetic will locate readability and perception into the topic during this textbook.
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Additional resources for An Introduction to Tensors and Group Theory for Physicists
Furthermore, if we have two linear operators A and B and we define their product (or composition) AB as the linear operator (AB)(v) ≡ A B(v) , you can then show that [AB] = [A][B]. Thus, composition of operators becomes matrix multiplication of the corresponding matrices. 8 For two linear operators A and B on a vector space V , show that [AB] = [A][B] in any basis. 13 Now con2 2 ∂ ∂ − y ∂x ) on this space. sider the familiar angular momentum operator Lz = −i(x ∂y You can check that 13 We have again ignored the overall normalization of the spherical harmonics to avoid unnecessary clutter.
7 Suppose T (v) = 0 ⇒ v = 0. Show that this is equivalent to T being oneto-one, which by the previous exercise is equivalent to T being one-to-one and onto, which is then equivalent to T being invertible. An important point to keep in mind is that a linear operator is not the same thing as a matrix; just as with vectors, the identification can only be made once a basis is chosen. ,n . 13) i,j =1 where the numbers Ti j , again called the components of T with respect to B,12 are defined by 11 Throughout this text I will denote the identity operator or identity matrix; it will be clear from context which is meant.
11 Let (· | ·) be an inner product. If a set of non-zero vectors e1 , . . e. (ei |ej ) = 0 when i = j , show that they are linearly independent. e. (ei |ej ) = ±δij ) is just an orthogonal set in which the vectors have unit length. 17 The dot product (or Euclidean metric) on Rn Let v = (v 1 , . . , v n ), w = (w 1 , . . , w n ) ∈ Rn . Define (· | ·) on Rn by n (v|w) ≡ v i wi . i=1 This is sometimes written as v · w. 7 is an orthonormal basis. 18 The Hermitian scalar product on Cn Let v = (v 1 , .