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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MICROBIOLOGICAL ENGINEERING - QUESTION 28.1 : In the calculation of the growth of bacteria, colony forming unit (CFU) in serial dilution is used. In a laboratory, viable count assay is used to estimate CFU. Formula applied is CFU / mL = (number of colonies x dilution) / (amount plated, in unit mL). Acceptable plate count is either between 20 and 200 or between 30 and 300 according to 2 different references. A wastewater sample of 200 ml is added to and mixed with 1.8 L of sterile water. Another 200 ml of the mixture is added to and mixed with 1.8 L of sterile water. (a) Calculate the dilution of first mixture and the dilution of the second mixture. (b) 100 microlitres of wastewater samples from the first mixture and the second mixture are placed separately on 2 different alga plates. The first plate has 250 colonies and the second plate has 23 colonies. Calculate the average CFU / mL.
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