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Transcript
763313A QUANTUM MECHANICS II Exercise 1 spring 2017 1. Let A and B be two matrices. a) Assume that A and B are such matrices that the product AB is well-defined. Show that (AB)† = B † A† . b) Show that for 2 × 2 matrices det(AB) = det(A) det(B). Does this result hold for all square matrices? 2. Calculate the eigenvalues and eigenvectors of the Pauli spin matrices σ1 = 0 1 1 0 , σ2 = 0 −i i 0 , σ3 = 1 0 . 0 −1 Show that the Pauli matrices are hermitian. 3. Calculate the expectation values hα|σi |αi, i = 1, 2, for Pauli spin matrices σ1 and σ2 with respect to an arbitrary state |αi = α1 |1i + α2 |2i. Here |1i and |2i are the eigenvectors of σ3 . 4. Consider a three-dimensional vector space spanned by an orthonormal basis |1i, |2i, |3i. Kets |αi and |βi are given by |αi = i|1i − 2|2i − i|3i, |βi = i|1i + 2|3i. Find all nine matrix elements of the operator  = |αihβ|, in this basis, and construct the matrix representing Â. Is it hermitian? 5. The Hamiltonian for a certain two-level system is Ĥ = (|1ih1| − |2ih2| + |1ih2| + |2ih1|), where |1i, |2i is an orthonormal basis and is a number with the dimensions of energy. Find its eigenvalues and eigenvectors (as linear combinations of |1i and |2i). What is the matrix representing Ĥ with respect to this basis?