University:
Widener University
Course:
PHYS-121 | Fundamentals of Physics I
Academic year:

2023
Views:

70

Pages:

2
Author:

eskaleerz0v

Problem Set #1 1. See Problem Set # 1, 1977, Problem 1. 2. See Problem Set # 1, 1977, Problem 3. 3. See Problem Set # 1, 1977, Problem 4. 4. You are going to use the Van Vleck Transformation (a fancy name for second-order quasi-degenerate perturbation theory) to solve a coupled harmonic oscillator problem. Consider H = H◦x + H◦y + ζ x2 y2 and let ω x = ωy . A. Given that � � � n x |x2 |n x = (n x + 1/2) mω � � �1/2 � � 2 n x |x2 |n x ± 2 = n x + n x + 1 ± (2n x + 1) mω ζ� ≡b 4m2 ω3 Construct the H matrix through n = n x + ny = 4. Notice that the matrix factors nicely into an odd n and an even n block. B. Apply the Van Vleck transformation to the n = 0, 1, and 2 blocks. Assume that �ω � b. Be sure to include corrections due to oﬀ-diagonal elements with n > 4 basis functions. Your bookkeeping will be simpliﬁed if you make use of “railroad” diagrams. For example, √ 3 6b 1,1 � � � � � � 3,1 √ 3 6b 6b √ 3 6b √ 1,3 3,3 3 6b 6b � � � 1,1 � � � Page 2 The diagram places the initial and ﬁnal basis functions at the left and at the right and in the middle are all basis functions that can simultaneously have non-zero matrix elements with both. Above each line is the actual value of the matrix element. All that remains is to look up the relevant energy denominators. The correction to the 1, 1; 1, 1 matrix element is (neglecting b terms in denominators) 54b2 54b2 36b2 − − − . 2�ω 2�ω 4�ω C. Now that you have eﬀectively uncoupled the n = 0, 1, 2 blocks from all other blocks, you can focus your attention individually on these 1 ×1, 2×2, and 3×3 isolated eﬀective Hamiltonians. Construct the (2, 0) ± (0, 2) and (1, 0) ± (0, 1) basis functions. This is called a “Wang Transfor mation”. Can you suggest any physical basis for this additional factorization? You should ﬁnd that the n = 0 through 2 part of this Hamiltonian is now fully diagonal. Draw an energy level diagram for n = 0 through 2 which compares your eigenvalues against the unperturbed (b = 0) levels.