Solutions to Physics 622 Homework 8: Exercises 3-11, Assignments of Quantum Mechanics

Download Solutions to Physics 622 Homework 8: Exercises 3-11 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 622 Spring 2000 - Dr. D. Fivel Homework 8 - Solutions Chapter 8 - Problems 1-12 Note: For problems 1,2,7, 9a,9d,10c,and 12 which only require straight forward substitution no solution is given below. √ Exercise 3a: Clearly L2f(X2) = f(X2)L2 since X2 is a rotational invariant. The assertion is then immediate. √ Exercise 3b: The one dimensional representation is multiplication by the number 1. √ Exercise 4: if A1, A2, · · · , AN are hermitian then (A1A2 · · ·AN )† = ANAN−1 · · ·A1. Hence the product is hermitian if and only if it is a palindrome, i.e. is unchanged when written backwards (like the word RADAR). The two terms in 4.21 d are palindromes. √ Exercise 5: ∑ j KjX 2Kj = ∑ j [Kj , X2]Kj +X2K2 = −2i ∑ j XjKj +X2K2 = −2iX ·K +X2K2 K ·X = n∑ j=1 [Kj , Xj ] +X ·K = −ni+X ·K The assertion then follows by substitution. √ Exercise 6: iX ·K “ = ” ∑ j xj ∂ ∂xj . But r ∂ ∂r = r ∑ j ∂xj ∂r ∂ ∂xj . Since xj = r fj(θ, φ) where fj is a function only of the angles, it follows that ∂xj/∂r = xj/r, and the assertion of 4.24a follows. The other parts follow by using this result. √ Exercise 8 : In an earlier homework you showed that P anti-commutes with X and K whereas T commutes with X and anti-commutes with K. Since L = X ∧K it follows that P commutes with L and T anticommutes with L It then follows that T |,m〉 is an eigenstate of L3 with the reverse eigenvalue whereas P|,m〉 is an eigenstate of L3 with HW-8 - 1