p1 - dimensions act on fermions in nature for example on...

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Problem Set 1 Physics 341 Due October 15 Some abbreviations: S - Shankar. 1 . If two Hermitian operators A and B commute, show that they can be simultaneously diagonalized. 2 . Write down the most general 2 × 2 unitary matrix. Find the eigenvalues of this matrix. 3 . Prove the Schwarz inequality; namely that for any two vectors | v i , | w i in an inner product, the following relation is true: || v || · || w || ≥ |h v | w i| . 4 . We claimed in lecture (but did not prove) that a matrix can be diagonalized by a unitary similarity transformation iff it is normal. A normal matrix satisfies [ M, M ] = 0 . Show that this claim is true. 5 . The kernel of a linear transformation T : V W consists of all vectors | v i ∈ V such that T | v i = | ~ 0 i . Show that the kernel is itself a vector space. 6 . The Pauli matrices are a beautiful and fundamental set of matrices in physics. The reason for their importance is that they are used to describe how spatial rotations in three
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Unformatted text preview: dimensions act on fermions in nature; for example, on electrons. (i) These matrices are given by σ 1 = ± 0 1 1 0 ² , σ 2 = ±-i i ² , σ 3 = ± 1-1 ² . Compute the commutator [ σ i ,σ j ] and the anti-commutator { σ i ,σ j } . (ii) Add the identity matrix σ = 1 to these three matrices and show that any 2 × 2 matrix M can be written in the form M = 3 X i =0 α i σ i for some complex α i . Under what conditions on α i is M unitary? Under what conditions is M Hermitian? 1 (iii) Consider the operator P = p σ + X i p i σ i where the 4-momentum p = ( p ,p 1 ,p 2 ,p 3 ) is real. What is det( P )? What are the eigenval-ues and eigenvectors of P ? 2...
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