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homework2

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Homework #2 — PHYS 402 — Spring 2011 Deadline: Wednesday, February 16, 2011, in class Professor Victor Yakovenko Office: 2115 Physics Web page: http://www2.physics.umd.edu/ ~ yakovenk/teaching/ Textbook: David J. Griffiths, Introduction to Quantum Mechanics 2nd edition, Pearson Prentice Hall, 2005, ISBN 0-13-111892-7 Do not forget to write your name and the homework number! Be sure to answer all additional questions asked in this assignment! Total score is 40 points. Ch. 4 Quantum Mechanics in Three Dimensions 1. Problem 4.26, 6 points. Properties of the Pauli matrices. Additional part, 2 points: Using Eq. (4.153) derived in part (b), show that the commutator of the Pauli matrices is [ σ j , σ k ] = σ j σ k - σ k σ j = 2 i X l jkl σ l , and the anticommutator is { σ j , σ k } = σ j σ k + σ k σ j = 2 δ jk . Notation here is the same as in part (b) of the problem. 2. Problem 4.28, 6 points. Expectation values for the most general spinor. Keep in mind that the spinor components a and b are complex numbers. 3. Problem 4.29, 6 points Eigenvectors and eigenvalues of ˆ S y . Follow the method described on page 175.

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Unformatted text preview: Additional part, 2 points: Answer question (b) of this problem also for the wave function shown in Example 4.2 on page 175. 4. Problem 4.49, 8 points. Do this problem for the following spinor χ = A 2 + i 1-i ! , not for the spinor given in the textbook. 5. Problem 4.31, 6 points. Spin matrices for spin 1. Hints: The matrix for ˆ S z can be easily constructed similarly to Eqs. (4.144) and (4.145). Then, use Eq. (4.136) for s = 1 and construct the operators ˆ S + and ˆ S-similarly to Eq. (4.146). Then, obtain ˆ S x and ˆ S y similarly to Eq. (4.147). See next page 2 Homework #2, Phys402, Spring 2011, Prof. Yakovenko Additional part, 4 points: Consider a general wave function for spin 1: χ = A a b c , where a , b , and c are some complex numbers. Determine the normalization coeﬃcient A and the expectation values of ˆ S x , ˆ S y , and ˆ S z in this state. February 9, 2011...
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