Question

At a given time t, a diatomic rigid rotor is found in a mixed quantum state describe by the function: where Ym are the normalized spherical harmonics, and N is the normalization constant. a) Normalize the function. (b) Compute the probability that a single measurement of the L-component in this quantum state can produce the result (La)classical . (c) Compute the mean rotational energy for one mole of 1C)'S molecules found in this rotational state. [The equilibrium bond length of CS is 1.519A.]

Answer 1

Psi Vertical-bar theta, phi Vertical-bar = N[1/30 + 1/20 + 1/2 1/1, 1 + 1/2 1/1, -1] Vertical-bar theta, phi > = N [ Vertical-bar 3, 0 > + Vertical-bar 2, 0 > + 1/2 Vertical-bar 1, 1 > + 1/2 Vertical-bar 1, -1 >] (a) Normalisation, 1 = < theta, phi Vertical-bar theta, phi > = Vertical-bar N Vertical-bar ^2 [< 3, 0 Vertical-bar 3, 0 > + < 2, 0 Vertical-bar 2, 0 > + 1/4 < 1, 1 Vertical-bar 1, 1 > + 1/4 < 1, -1 Vertical-bar 1, -1 >] 1 = Vertical-bar N Vertical-bar ^2 [1 + 1 + 1/4 + 1/4] = Vertical-bar N Vertical-bar ^2 [2 + 2/4] = Vertical-bar N Vertical-bar ^2 (8 + 2/4) 1 = Vertical-bar N Vertical-bar ^2 times 105/4 N = squareroot 2/5 Normalized state Vertical-bar theta, phi > = squareroot 2/5 [ Vertical-bar 3, 0 > + Vertical-bar 2, 0 > + 1/2 Vertical-bar 1, 1 > + 1/2 Vertical-bar 1, -1 >] Vertical-bar theta, phi > = squareroot 2/5 Vertical-bar 3, 0 > + squareroot 2/5 Vertical-bar 2, 0 > + 1/squareroot 10 Vertical-bar 1, 1 > + 1/squareroot 10 Vertical-bar 1, -1 > (b) Probability of getting _z = 0h I^st for state Vertical-bar 3, 0 > P_1 = Vertical-bar < 3, 0 Vertical-bar theta, phi Vertical-bar ^2