4604_ProblemSet4_fa07

# 4604_ProblemSet4_fa07 - PHY4604 Fall 2007 Problem Set 4 PHY...

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PHY4604 Fall 2007 Problem Set 4 Department of Physics Page 1 of 3 PHY 4604 Problem Set #4 Due Wednesday October 17, 2007 (in class) (Total Points = 110, Late homework = 50%) Reading: Griffiths Chapter 3 and the Appendix. Useful Integrals: π = +∞ dx x x 2 2 ) ( sin . Problem 1 (25 points): . (a) (2 points) Show that the sum of two hermitian operators is hermitian (b) (2 points) Suppose that H op is a hermitian operator, and α is a complex number. Under what condition (on α ) is α H op hermitian? (c) (2 points) When is the product of two hermitian operators hermitian? (d) (2 points) If dx d O op = , what is op O ? (e) (2 points) Show that = op op op op A B B A ) (. (f) (2 points) Prove that [AB,C] = A[B,C] + [A,C]B, where A, B, and C are operators. (g) (2 points) Show that dx df i x f p op x h = )] ( , ) [( , for any function f(x). (h) (2 points) Show that the anti-hermitian operator, I op , has at most one real eigenvalue (Note: anti-hermitian means that op op I I = ). (i) (2 points) If A op is an hermitian operator, show that 0 2 >≥ < op A . (j) (2 points) The parity operator, P op , is defined by P op Ψ (x,t) = Ψ (-x,t). Prove that the parity operator is hermitian and show that 1 2 = op P , where 1 is the identity operator.

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4604_ProblemSet4_fa07 - PHY4604 Fall 2007 Problem Set 4 PHY...

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