5.73 Quiz 17 ANSWERS 1. = φv1 (x1 )φv2 (x 2 ) ψv(0) 1v 2 E v(0) = ! [ ω1 (v1 + 1 / 2) + ω 2 (v 2 + 1 / 2)] 1v 2 H(1) = k122 x1x 22 ⎛mω ⎞ x1 = ⎜ 1 1 ⎟ ⎝ 2! ⎠ 1/2 ⎛m ω ⎞ x2 = ⎜ 2 2 ⎟ ⎝ 2! ⎠ A. n2 x 22 k 2 H(1) n1k1n2 k 2 = k122 n1 x1 k1 (a 1/2 1 (a + a1† ) 2 + a †2 ) Matrix elements of H have four indices, n and k specify the final and initial state quantum numbers for oscillator #1, n and k the final and initial state quantum numbers for oscillator #2. What are the selection rules for nonzero matrix elements of H ? n – k = ±1 (1) 1 1 2 2 (1) 1 1 n – k = 0, ±2 2 B. 2 What are the zero-order energy differences that correspond to each of the nonzero matrix elements of H(1) n1k1n2 k 2 ? ( ) ( ) E n(0) − E k(0) = ! ⎡ω1 n1 − k1 + ω 2 n2 − k 2 ⎤ 1n2 1k 2 ⎣ ⎦ 1 C. Evaluate at least two of the six nonzero values of the off-diagonal elements of H(1). (i) e.g. Hn1n1 +1n2n2 +2 = γ ⎡⎣( n1 + 1)( n2 + 2 )( n2 + 1) ⎤⎦ 1/2 DO THIS ONE! (ii) 1/2 Hn1n1 +1n2n2 = γ ⎡⎢ ( n1 + 1) ( 2 n2 + 1) ⎤⎥ ⎦ ⎣ (iii) Hn1n1 +1n2n2 −2 = γ ⎡ ( n1 + 1)( n2 )( n2 − 1) ⎤ ⎣ ⎦ (iv) Hn1n1 −1n2n2 +2 = γ ⎡ n1 ( n2 + 2 )( n2 + 1) ⎤ ⎣ ⎦ (v) Hn1n1 −1n2n2 = γ ⎡ n11/2 ( 2 n2 + 1) ⎤ ⎣ ⎦ (vi) Hn1n1 −1n2n2 −2 = γ ⎡ n1n2 ( n2 − 1) ⎤ ⎣ ⎦ 1/2 1/2 1/2 ⎛mω ⎞ where γ ≡ ⎜ 1 1 ⎟ ⎝ 2! ⎠ D. The term 1/2 ⎛ m2ω 2 ⎞ ⎜⎝ ⎟ . 2! ⎠ H(1) n1n1 +1n2 n2 −2 1 2 E n(0) − E n(0) 1n2 1 +1n2 −2 appears in the second-order perturbation summation for E n(2) . 1n2 Evaluate this term (based on your answers to parts B and C(iii)). γ 2 ( n1 + 1)( n2 )( n2 − 1) ! ( 2ω 2 − ω1 ) E. What happens to the term in part D if ω1 = 2ω2? Blows up! 2