HW1 - 1 = 3 1 , 2 = 1 2 , 3 = 1 1 3 . 7. Let V be the space...

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Physics 731 Assignment #1, due Wednesday, September 16 1. Sakurai, Problem 2, Chapter 1. 2. Sakurai, Problem 3, Chapter 1. 3. Determine the values of the real number x such that x 2 1 , 2 x - 1 2 , 3 4 2 are linearly independent. 4. Express the vector β = 4 2 1 in terms of the basis vectors α 1 = 1 1 1 , α 2 = 1 0 - 1 , α 3 = 0 - 1 2 . 5. Show that the following vectors form an orthonormal basis α 1 = 1 6 i + 1 i - 1 - i - 1 , α 2 = 1 2 0 i 1 , α 3 = 1 6 2 i 1 i , and represent β 1 = 1 1 1 , β 2 = - 3 i - 2 in terms of the α i . 6. Use the Gram-Schmidt procedure to construct an orthonormal basis starting from the vectors
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Unformatted text preview: 1 = 3 1 , 2 = 1 2 , 3 = 1 1 3 . 7. Let V be the space of real polynomials dened on the interval [-1 , 1] . Dene the inner product h P | Q i = R 1-1 P ( x ) Q ( x ) dx , and the non-orthonormal basis R i ( x ) = x i , for i = 0 , . . . . Use the Gram-Schmidt procedure to construct the rst three vectors in an orthonormal basis. 8. Sakurai, Problem 14, Chapter 1. Recall 1 , 2 , 3 are the Pauli matrices. 1...
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