QUANTUM CHEMISTRY AND CHEMICAL ENGINEERING - EXAMPLE 31.7 : (a) The correct statement about both the average value of position (<x>) and momentum (<p>) of a 1-dimensional harmonic oscillator wavefunction is <x> = <p> = 1 - x. Find the value of x. (b) The probabilities of finding a particle around points A, B and C in the wavefunction y = f(x) are P(A), P(B) and P(C) respectively. Coordinates are A (3,5), B (4,-10) and C (6,7). Arrange P(A), P(B) and P(C) in term of a < b < c, when | y-coordinate | signifies the probability.
QUANTUM CHEMISTRY AND CHEMICAL ENGINEERING - ANSWER 31.7 : (a) Let <x> = <p> = 1 - x = 0, then x = 1 - 0 = 1. (b) With reference to the | y-coordinate | for each point in A, B and C coordinates, if 5 < 7 < |-10| then a < b < c = P(A) < P(C) < P(B). 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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Question 102 - (a) As an approximation, let v = Zc / 137 where v is the radial velocity for 1 s electron of an element, c is the speed of light, Z is the atomic number. For gold with Z = 79, find the radial velocity of its 1 s electron, in term of c and percentage of the speed of light. (b) As an approximation, let A x A = 1 - Z x Z / 18769 where A is the ratio of the relativistic and non-relativistic Bohr radius. Find the value of A.
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Question 33 – By using Excel program either on laptop or desktop PC, solve the differential equation dy / dx = -2y + x + 4 with h = 0.005, initial values : x = 0, y = 1. The 4th order Runge-Kutta method provides : y(N + 1) = y(N) + (1/6) (k1 + 2k2 +2k3 + k4), k1 = h [ -2y(N) + x(N) + 4 ], k2 = h { -2 [ y(N) + k1 / 2 ] + x(N) + h / 2 + 4 }, k3 = h { -2 [ y(N) + k2 / 2 ] + x(N) + h / 2 + 4 }, k4 = h { -2 [ y(N) + k3 ] + x(N) + h + 4 }. What is the value of y at x = 0.5?
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CHEMICAL FLUID MECHANIC - EXAMPLE 3.3 : The drag coefficient Cd = 0.05 and lift coefficient Cl = 0.4 for a levelled flow aircraft are measured. The velocity of the aircraft is v = 150 ft / s with its weight W = 2677.5 pound-force. (a) Find the value of the lift of the aircraft, L, when it is also its weight. (b) The drag of the aircraft, D = Cd M, L = Cl M. Find the value of D. (c) The power required is P = Dv. If 1 pound-force x (ft / s) = 1.356 W, find the value of P in the unit of Watt or W.
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QUANTUM COMPUTING - EXAMPLE 32.2 : (a) If | 001 > = | 1 >, | 111 > = | 7 >, find the 2 possible values of ( | 001 > + | 1 > + | 7 > ) ( | 111 > ). (b) In quantum money, a duplicate will have probability P of passing the verification test of a bank, if the total number of photons on the bank note is N. The would be counterfeiter has a probability p of success in duplicating the quantum money correctly for each photon. Guess the relationship of P, p and N as a mathematical formula involving natural logarithm ln.
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