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1. Consider the following operators: a. T = ve dax2 b. p = Wild c. 8=...
What are the results of operating on the following functions with the operators d/dx and d2/dx2...
What are the results of operating on the following functions with the operators d/dx and d2/dx2 : a) 4x-3 b) cos(bx), c) eikx d) (x2–i) ? What functions are eigenfunctions of these operators? What are the corresponding eigenvalues?
Four wave functions are given below. III. IV. yx, t) = 5sin(4x - 20t+4) y(x, t)...
Four wave functions are given below. III. IV. yx, t) = 5sin(4x - 20t+4) y(x, t) = 5sin(3x – 12t+5) y(x, t) = 5cos(4x + 24 + 6) y(x,0) = 14cos(2x - 8t+3) Use this exhibit to answer the following question(s). Refer to Exhibit 16-3. Rank the wave functions in order of the magnitude of the frequencies of the waves, from least to greatest. O a. III, IV, II, I b. IV = II, I, III C. IV. I, II,...
Evaluate the following integrals (from A to E) A. Integration by parts i) ſ (3+ ++2)...
Evaluate the following integrals (from A to E) A. Integration by parts i) ſ (3+ ++2) sin(2t) dt ii) Z dz un (ricos x?cos 4x dx wja iv) (2 + 5x)eš dr. B. Involving Trigonometric functions 271 п i) | sin? ({x)cos*(xx) dx ii) Sco -> (=w) sins (įw) iii) sec iv) ſ tan” (63)sec^® (6x) dx . sec" (3y)tan?(3y)dy C. Involving Partial fractions 4 z? + 2z + 3 1) $77 dx 10 S2-6922+4) dz x2 + 5x -...
qm 09.4 4. The commutation relations defining the angular momentum operators can be written [Îx, Îy]...
qm 09.4
4. The commutation relations defining the angular momentum operators can be written [Îx, Îy] = iħẢz, with similar equations for cyclic permutations of x, y and z. Angular momentum raising and lowering operators can be defined as În = Îx ihy (i) Show that [Lz, L.] = +ħL. [6 marks] (ii) If øm is an eigenfunction of ł, with eigenvalue mħ, show that the state given by L+øm is also an eigenfunction of L, but with an eigenvalue...
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i....
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii. Find the corresponding eigenvalue relation and eigenfunctions. Note that you do not have to normalise the eigenfunctions. (b) Solve the heat equation (2) 0<х<1 t>0 Ut u(0, t) u(1,t) u(x, 0) sin(2тx) + 1
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii....
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B =...
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [
3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ]
(i) Calculate A + B,
(ii) Calculate AB
(iii) Calculate the inverse of B,
(iv) Calculate the determinant of C.
(b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2)
and (5, −1, 4) respectively.
(i) Show that P Q~ =...
T-1 Suppose two events A and B are mutually exclusive and PAI 0, P[B] 0 ....
T-1 Suppose two events A and B are mutually exclusive and PAI 0, P[B] 0 . Consider the following statements: i) P(An B)=0 ii) P(A U B) = P(A) + P(B) iii) A and B are statistically independent. Choose the correct statement. A) Only i) is true. B) Only ii) is true. C) Only iii is true. D) Only i) and i) are true. E) i), ii) and iii) are all true.
Consider the below wave function and answer the following questions. F(t) cos(T.6t)+cos(.5t) i) Graph the beat...
Consider the below wave function and answer the following questions. F(t) cos(T.6t)+cos(.5t) i) Graph the beat function for this wave. ii) What is the beat frequency? iii Calculate the proper sampling rate for this wave. iv) Calculate the time intervals between the samples.
Consider the below wave function and answer the following questions. F(t) cos(T.6t)+cos(.5t) i) Graph the beat function for this wave. ii) What is the beat frequency? iii Calculate the proper sampling rate for this wave. iv) Calculate...
Match each of the following functions, f, with p: its (truncated) Taylor polynomial approx- imation about...
Match each of the following functions, f, with p: its (truncated) Taylor polynomial approx- imation about a = 0. (d) f(x) = V4+. (a) f(x) = {x?; (b) f(x) = e-62; (@) f(x) = cos(2x); (1) p(x) = 1 – 62 +18x2 – 3624 – 32470. (i) P(x) = 2 + - 64 x2 + 5127 (ii) P(x) = 1 – 2x2 + xy - 1 10. (iv) p(x) = 1 +2° +2° +
For each of the following functions, (i) find the constant c so that f(x) is a...
For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable x, (ii) find the cdf F(x)-P(XSX), (iii) sketch graphs of the pdf f (x) and the distribution function F(x), and (iv) find μ and σ2. (a) f (x) x3/4, 0 <x<c (b) f (x)-(3/16x-,-c < x c
Four wave functions are given below. III. IV. yx, t) = 5sin(4x - 20t+4) y(x, t) = 5sin(3x – 12t+5) y(x, t) = 5cos(4x + 24 + 6) y(x,0) = 14cos(2x - 8t+3) Use this exhibit to answer the following question(s). Refer to Exhibit 16-3. Rank the wave functions in order of the magnitude of the frequencies of the waves, from least to greatest. O a. III, IV, II, I b. IV = II, I, III C. IV. I, II,...
Evaluate the following integrals (from A to E) A. Integration by parts i) ſ (3+ ++2) sin(2t) dt ii) Z dz un (ricos x?cos 4x dx wja iv) (2 + 5x)eš dr. B. Involving Trigonometric functions 271 п i) | sin? ({x)cos*(xx) dx ii) Sco -> (=w) sins (įw) iii) sec iv) ſ tan” (63)sec^® (6x) dx . sec" (3y)tan?(3y)dy C. Involving Partial fractions 4 z? + 2z + 3 1) $77 dx 10 S2-6922+4) dz x2 + 5x -...
qm 09.4
4. The commutation relations defining the angular momentum operators can be written [Îx, Îy] = iħẢz, with similar equations for cyclic permutations of x, y and z. Angular momentum raising and lowering operators can be defined as În = Îx ihy (i) Show that [Lz, L.] = +ħL. [6 marks] (ii) If øm is an eigenfunction of ł, with eigenvalue mħ, show that the state given by L+øm is also an eigenfunction of L, but with an eigenvalue...
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii. Find the corresponding eigenvalue relation and eigenfunctions. Note that you do not have to normalise the eigenfunctions. (b) Solve the heat equation (2) 0<х<1 t>0 Ut u(0, t) u(1,t) u(x, 0) sin(2тx) + 1
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii....
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [
3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ]
(i) Calculate A + B,
(ii) Calculate AB
(iii) Calculate the inverse of B,
(iv) Calculate the determinant of C.
(b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2)
and (5, −1, 4) respectively.
(i) Show that P Q~ =...
T-1 Suppose two events A and B are mutually exclusive and PAI 0, P[B] 0 . Consider the following statements: i) P(An B)=0 ii) P(A U B) = P(A) + P(B) iii) A and B are statistically independent. Choose the correct statement. A) Only i) is true. B) Only ii) is true. C) Only iii is true. D) Only i) and i) are true. E) i), ii) and iii) are all true.
Consider the below wave function and answer the following questions. F(t) cos(T.6t)+cos(.5t) i) Graph the beat function for this wave. ii) What is the beat frequency? iii Calculate the proper sampling rate for this wave. iv) Calculate the time intervals between the samples.
Consider the below wave function and answer the following questions. F(t) cos(T.6t)+cos(.5t) i) Graph the beat function for this wave. ii) What is the beat frequency? iii Calculate the proper sampling rate for this wave. iv) Calculate...
Match each of the following functions, f, with p: its (truncated) Taylor polynomial approx- imation about a = 0. (d) f(x) = V4+. (a) f(x) = {x?; (b) f(x) = e-62; (@) f(x) = cos(2x); (1) p(x) = 1 – 62 +18x2 – 3624 – 32470. (i) P(x) = 2 + - 64 x2 + 5127 (ii) P(x) = 1 – 2x2 + xy - 1 10. (iv) p(x) = 1 +2° +2° +
For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable x, (ii) find the cdf F(x)-P(XSX), (iii) sketch graphs of the pdf f (x) and the distribution function F(x), and (iv) find μ and σ2. (a) f (x) x3/4, 0 <x<c (b) f (x)-(3/16x-,-c < x c
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