New PDF release: An Introduction to Tensors and Group Theory for Physicists

By Nadir Jeevanjee

ISBN-10: 0817647147

ISBN-13: 9780817647148

An creation 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 specific objective is to demystify tensors and supply a unified framework for figuring out them within the context of classical and quantum physics. Connecting the part formalism standard in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings should be a welcome boost 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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Example text

If one considers the operator ∞ exp(−iφLz ) ≡ n=0 1 (−iφLz )n n! 1 1 (−iφLz )2 + (−iφLz )3 + · · · 2! 3! (the usual power series expansion for ex ) then one gets the operator which represents a rotation about the z axis by an angle φ. 15 to get = I − iφLz + 21 See Sakurai [14] or Gasiorowicz [5] or our discussion in Chap. 4. 8 Problems 35 ⎛ exp −iφ[Lz ]{x,y,z} − sin φ cos φ 0 cos φ = ⎝ sin φ 0 ⎞ 0 0⎠, 1 the usual matrix for a rotation about the z-axis. 4 Let V be finite-dimensional, and consider the ‘double dual’ space (V ∗ )∗ .

You have doubtless seen many examples of linear operators: for instance, we can interpret a real n × n matrix as a linear operator on Rn that acts on column vectors by matrix multiplication. Thus Mn (R) (and, similarly, Mn (C)) can be viewed as vector spaces whose elements are themselves linear operators. 4; in that case, we identified elements of H2 (C) as the quantum-mechanical angular momentum operators. There are numerous other examples of quantum-mechanical linear operators: for instance, the familiar position and momentum operators xˆ and pˆ that act on L2 ([−a, a]) by (xf ˆ )(x) = xf (x) ∂f (pf ˆ )(x) = −i (x) ∂x as well as the angular momentum operators Lx , Ly , and Lz which act on Pl (R3 ) by ∂f ∂f −z ∂z ∂y ∂f ∂f −x Ly (f ) = −i z ∂x ∂z ∂f ∂f −y .

Ejs ≡ v1i1 . . vrir f1j1 . . js ≡ T ei1 , . . , eir , ej1 , . . ,n . You should check that this definition of the components of a tensor, when applied to vectors, dual vectors, and linear operators, agrees with the definitions given earlier. 5) gives us a concrete way to think about the components of tensors: they are the values of the tensor on the basis vectors. 1 By choosing suitable definitions of addition and scalar multiplication, show that Tsr (V ) is a vector space. If we have a non-degenerate bilinear form on V , then we may change the type of T by precomposing with the map L or L−1 .

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An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee


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