ENGINEERING PHYSICS - EXAMPLE 30.4 : (a) Time evolution in Heisenberg picture, according to Ehrenfest theorem is m (d / dt) < r > = < p >, where m = mass, r = position, p = momentum of a particle. If v = velocity, prove that m < v > = < p >. (b) Lande g-factor is given by Gj = Gl [ J (J + 1) - S (S + 1) + L (L + 1) ] / [ 2J (J + 1) ] + Gs [ J (J + 1) + S (S + 1) - L (L + 1) ] / [ 2J (J + 1) ]. If Gl = 1 and under approximation of Gs = 2, prove by calculation that Gj = (3/2) + [ S (S + 1) - L (L + 1) ] / [ 2J (J + 1) ].
ENGINEERING PHYSICS - ANSWER 30.4 : (a) Let (d / dt) < r > = < v >. Substitute it into m (d / dt) < r > = < p > so that m < v > = < p >, proven by momentum = mass x velocity. (b) Gj = Gl [ J (J + 1) - S (S + 1) + L (L + 1) ] / [ 2J (J + 1) ] + Gs [ J (J + 1) + S (S + 1) - L (L + 1) ] / [ 2J (J + 1) ]. When Gl = 1 and Gs = 2, Gj = [ J (J + 1) - S (S + 1) + L (L + 1) ] / [ 2J (J + 1) ] + 2 [ J (J + 1) + S (S + 1) - L (L + 1) ] / [ 2J (J + 1) ] = [ J (J + 1) + 2J (J + 1) ] / [ 2J (J + 1) ] + [ 2S (S + 1) - 2L (L + 1) - S (S + 1) + L (L + 1) ] / [ 2J (J + 1) ] = [ 3J (J + 1) ] / [ 2J (J + 1) ] + [ S (S + 1) - L (L + 1) ] / [ 2J (J + 1) ] = (3/2) + [ S (S + 1) - L (L + 1) ] / [ 2J (J + 1) ] (proven). The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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ENVIRONMENTAL ENGINEERING - QUESTION 22.1 : In order to predict the wastewater production, the population number has to be understood. The population data is : 72000 (for year 1961 or P-1961), 85000 (for year 1971 or P-1971), 110500 (for year 1981 or P-1981). (a) Find the average population increase, or [ (P-1981 - P-1971) + (P-1971 - P-1961) ] / 2. (b) Find the average percentage population increase, or [ (P-1981 - P-1971) / P-1971 + (P-1971 - P-1961) / P-1961 ] / (2) X 100. (c) Find the incremental increase or P-1981 - 2 (P-1971) + P-1961. (d) Let Po = P-1981. After 2 decades or n = 2, the population is P-2001. By using arithmetical increase method, find P-2001 = Po + n (Answer for a). (e) By using incremental increase method, find P-2001 = (Answer of d) + n (n + 1) (Answer of c) / 2. (f) By using geometrical increase method, find P-2001 = Po [ 1 + (Answer of b) / 100 ] ^ n where ^ is power sign, or 1 ^ 2 = 1 x 1 = 1. (g) If the actual P-2001 = 184000, which method of estimation is more accurate, based on your answer in (d), (e) and (f)?
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