QUANTUM CHEMISTRY AND CHEMICAL ENGINEERING - EXAMPLE 31.6 : In N + 1 Rule in Quantum Chemistry, whenever a spin 1 / 2 nucleus is adjacent to N other nuclei, it is split into N + 1 distinct peaks. In 1 peak or singlet, there is only 1 magnitude. In 2 peaks or doublet, the ratio of magnitude of each peak is 1 : 1. In 3 peaks or triplet, the ratio of magnitude of each peak is 1 : 2 : 1. In 4 peaks or quartet, the ratio of magnitude of each peak is 1 : 3 : 3 : 1. In 5 peaks or quintet, the ratio of magnitude of each peak is 1 : 4 : 6 : 4 : 1. (a) By using binomial coefficients or Triangle of Pascal find the ratio of magnitude of each peak if 6 peaks exists. (b) How many adjacent nuclei are available in a spin 1 / 2 nucleus in such situation of 6 peaks?

QUANTUM CHEMISTRY AND CHEMICAL ENGINEERING - EXAMPLE 31.7 : (a) The correct statement about both the average value of position () and momentum (

) of a 1-dimensional harmonic oscillator wavefunction is =

= 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 - EXAMPLE 31.8 : (a) Acceptable wavefunction in quantum mechanics in the range of : negative infinity < x < positive infinity, vanishes at least at one boundary. Which of the following is the wavefunction or are the wavefunctions of acceptable theory : P = x, P = | x |, P = sin x, P = exp (-x), P = exp (-| x |)? State the reason. (b) Let linear momentum operator P = -ih d / dz. The wavefunction is S = exp (-ikz) where i x i = -1, k and h are constants. Find the linear momentum of such wavefunction by using the term P x S.

Question 112 - In quantum computing, let the amplitude A = a | 0 > + b | 1 >, | a | | a | + | b | | b | = 1. Find the values of b if A = 0.8 | 0 > + b | 1 >.

QUANTUM CHEMISTRY AND CHEMICAL ENGINEERING - EXAMPLE 31.9 : When an algebraic product is defined on the space, the Lie bracket is the commutator [x,y] = xy - yx according to Lie algebra in mathematics. If [p,x] f(x) = px f(x) - xp f(x), p = -ih d / dx, find the value of [p,x] in term of i and h.

QUANTUM CHEMISTRY AND CHEMICAL ENGINEERING - EXAMPLE 31.10 : There are 6 spin orbitals in p subshell in a ground state carbon atom. Only 2 electrons fill the p subshell. Number of different ways for n electrons to occupy the k spin orbitals are k! / [ (n!) (k-n)! ]. Find the number of different configurations of electrons to occupy the p subshell in a carbon atom.

QUANTUM COMPUTING - EXAMPLE 32.1 : In quantum computing, let the amplitude A = a | 0 > + b | 1 >, | a | | a | + | b | | b | = 1. Find the values of b if A = 0.8 | 0 > + b | 1 >.

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.

QUANTUM COMPUTING - EXAMPLE 32.3 : A system of linear congruences consists of 3 equations : X ≡ 1 (mod 3), X ≡ 3 (mod 5), X ≡ 4 (mod 6). X has positive values. (a) List the values of these equations from 1 to 35. Then find the minimum value of X. (b)(i) Find the least common multiple (LCM) of b = 3, 5 and 6 where X ≡ a (mod b). (ii) If b - a has the same value of all equations above, then X + (b - a) is divisible by LCM. Find the value of minimum value of X via LCM division.

QUANTUM COMPUTING - EXAMPLE 32.4 : A system of linear congruences consists of 3 equations : X ≡ 1 (mod 2), X ≡ 3 (mod 3), X ≡ 4 (mod 5). X has positive values. (a)(i) List the values of these equations from 1 to approximately 40. (ii) Find the first smallest value and second smallest value of X. (iii) Guess the third smallest value of X. (b) Let X ≡ Aa (mod Ma), X ≡ Ab (mod Mb), X ≡ Ac (mod Mc). According to Chinese remainder theorem, X ≡ (Aa x Ya x Md + Ab x Yb x Me + Ac x Yc x Mf) [ mod (Ma x Mb x Mc) ]. (i) Show that Ma, Mb and Mc have the greatest common divisor of Ma x Mb x Mc. (ii) Find the values of Md, Me and Mf if Md = Mb x Mc, Me = Ma x Mc and Mf = Ma x Mb. (iii) Find the values of Ya, Yb and Yc if Ya = Remainder of (Md / Ma), Yb = Remainder of (Me / Mb) and Yc = Remainder of (Mf / Mc). (iv) Use Chinese remainder theorem to find X.

QUANTUM COMPUTING - EXAMPLE 32.5 : In quantum teleportation, let (C0 + D1) (00 + 11) = C000 + C011 + D100 + D111. Ba = 00 + 11, Bc = 10 + 01, Be = 00 - 11, Bm = 10 - 01. (a) Find the values of 00, 01, 10 and 11 in term of Ba, Bc, Be and Bm. (b) Prove by calculation that (C0 + D1) (00 + 11) = 0.5 [ Ba (C0 + D1) + Bc (C1 + D0) + Be (C0 - D1) + Bm (-C1 + D0) ].

QUANTUM COMPUTING - EXAMPLE 32.6 : (a) Let H | 0 > = 0.707 ( | 0 > + | 1 > ), H | 1 > = 0.707 ( | 0 > - | 1 > ). Find the values for H | 0 > + H | 1 > and H | 0 > - H | 1 >. (b) In quantum computing, a qubyte is a quantum byte, or 8 quantum bits, a sequence processed as a unit. A qubit is a quantum bit. According to Alexander Holevo in his theorem, n qubits cannot carry more than n classical bits of information. What is the maximum amount of classical bits of information that can be carried by 1 qubyte.

QUANTUM COMPUTING - EXAMPLE 32.7 : If | ± > = 0.707 ( | 0 > ± | 1 > ), prove that | Ψ (t = 0) > = | 0 > = 0.707 ( | + > + | - > ).

QUANTUM COMPUTING - EXAMPLE 32.8 : In quantum computing, a quantum state is given by S = a | 00 > + b | 01 > + g | 10 > + d | 11 >. (a) Find S in term of | 0 > and | 1 > etc. (b) The probability of getting x is P(x). For S = 0.5 | 00 > + 0.5 | 01 > + 0.5 | 10 > + 0.5 | 11 >, find P(0) and P(1). Hint : P(00) + P(01) = P(0) = a x a + b x b, P(10) + P(11) = P(1) = g x g + d x d.

QUANTUM COMPUTING - EXAMPLE 32.9 : In quantum computing, find the equations of S = (a | 0 > + b | 1 >) (g | 0 > + d | 1 >) in term of | 00 >, | 01 >, | 10 > and / or | 11 > when ad = 0.